Mathematical issues concerning the Navier-Stokes equations and some of their generalizations By J. Málek1 and K. R. Rajagopal2 1 Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83, 186 75 Prague 8, Czech Republic 2 Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA This article primarily deals with internal, isothermal, unsteady flows of a class of incompressible fluids with both constant, and shear or pressure dependent viscosity that includes the Navier-Stokes fluid as a special subclass. We begin with a description of such fluids within the framework of a continuum. We then discuss various ways in which the response of a fluid can depart from that of a Navier-Stokes fluid. Next, we introduce a general thermodynamic framework that has been successful in describing the disparate response of continua that includes those of inelasticity, solid-to-solid transformation, viscoelasticity, granular materials, blood and asphalt rheology etc. Here, it leads to a novel derivation of the constitutive equation for the Cauchy stress for fluids with constant, or shear or pressure, or density dependent viscosity within a full thermo-mechanical setting. One advantage of this approach consists in a transparent treatment of the constraint of incompressibility. We then concentrate on mathematical analysis of three-dimensional unsteady flows of fluids with shear dependent viscosity that includes the Navier-Stokes model and Ladyshenskaya’s model as special cases. We are interested in the issues connected with mathematical self-consistency of the models, i.e., we are interested in knowing whether 1) flows exist for reasonable, but arbitrary initial data and for all instants of time, 2) flows are uniquely determined, 3) the velocity is bounded and 4) the large-time behavior of all possible $LaTeX: 2005/4/20 $ 2 J. Málek and K. R. Rajagopal flows can be captured by a finite dimensional, small (compact) set attracting all flow trajectories exponentially. For simplicity, we eliminate a choice of boundary conditions and their influence on flows assuming that all functions are spatially periodic with zero mean value over periodic cell. All results could be however extended to internal flows where the tangent component of the velocity satisfies Navier’s slip at the boundary. Most of the results hold also for no-slip boundary conditions. While the mathematical consistency understood in the above sense of the NavierStokes model in three dimension is not clear yet, we will show that Ladyzhenskaya’s model and some of its generalization enjoy all above properties for certain range of parameters. Briefly, we also discuss further results related to further generalizations of the Navier-Stokes equations. Keywords: incompressible fluid, Navier-Stokes fluid, non-Newtonian fluid, rheology, mathematical analysis The contribution of J. Málek to this work is a part of the research project MSM 0021620839 financed by MSMT. K. R. Rajagopal thanks the National Science Foundation for its support. A part of this research was performed during the stay of J. Málek at Department of Mathematics, Texas A&M University. The authors thank Miroslav Bulı́ček for his continuous help in the process of the preparation of this work, and Petr Kaplický for his critical comments to earlier drafts of this work. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 3 In memory of Olga Alexandrovna Ladyzhenskaya March 7, 1922 - January 12, 2004 and Jindřich Nečas December 14, 1929 - December 6, 2002 $LaTeX: 2005/4/20 $ 4 J. Málek and K. R. Rajagopal Chapter A Incompressible Fluids With Shear, Pressure and Density Dependent Viscosity from Point of View of Continuum Physics Contents 1. Introduction 7 1.1 What is a fluid? 7 1.2 Navier-Stokes fluid model 9 1.3 Departures From Newtonian Behavior 11 2. Balance equations 19 2.1 Kinematics 19 2.2 Balance of Mass - Incompressibility - Inhomogenity 21 2.3 Balance of Linear Momentum 22 2.4 Balance of Angular Momentum 23 2.5 Balance of Energy 23 2.6 Further Thermodynamic Considerations (The Second Law). Reduced dissipation equation 23 2.7 Isothermal flows at uniform temperature 26 2.8 Natural Configurations 27 3. The Constitutive Models For Compressible and Incompressible NavierStokes Fluids and Some of their Generalizations 28 3.1 Standard approach in continuum physics 28 3.2 Alternate approach 29 4. Boundary Conditions 31 $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 5 Chapter B Mathematical Analysis of Flows of Fluids With Shear, Pressure and Density Dependent Viscosity Contents 1. Introduction 35 1.1 A taxonomy of models 35 1.2 Mathematical self-consistency of the models 37 1.3 Weak solution: a natural notion of solution for PDEs of the continuum physics 38 1.4 Models and their invariance with respect to scaling 41 2. Definitions of (suitable) weak solutions 43 2.1 Assumptions concerning the stress tensor 43 2.2 Function spaces 44 2.3 Definition of Problem (P) and its (suitable) weak solutions 45 2.4 Useful inequalities 47 3. Existence of a (suitable) weak solution 49 3.1 Formulation of the results and bibliographical notes 3.2 Definition of an approximate Problem (P ε,η 49 ) and apriori estimates 51 3.3 Solvability of an approximative problem 54 3.4 Further uniform estimates w.r.t ε and η 56 3.5 Limit ε → 0 58 3.6 Limit η → 0, the case r ≥ 3.7 Limit η → 0, the case 8 5 11 5 <r< 60 11 5 3.8 Continuity w.r.t. time in weak topology of 61 L2per 65 3.9 (Local) Energy equality and inequality 66 3.10 Attainment of the initial condition 67 4. On smoothness of flows 67 4.1 A survey of regularity results 67 4.2 A cascade of inequalities for Ladyzhenskaya’s equations 72 4.3 Boundedness of the velocity 74 4.4 Fractional higher differentiability 74 $LaTeX: 2005/4/20 $ 6 J. Málek and K. R. Rajagopal 4.5 Short-time or small-data existence of ”smooth” solution 75 5. Uniqueness and large-data behavior 77 5.1 Uniquely determined flows described by Ladyzhenskaya’s equations 77 5.2 Large-time behavior - the method of trajectories 80 6. On structure of possible singularities for flows of Navier-Stokes fluid 83 7. Other incompressible fluid models 86 7.1 Fluids with pressure-dependent viscosity 86 7.2 Fluids with pressure and shear dependent viscosities 87 7.3 Inhomogeneous incompressible fluids 88 $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 7 Chapter A Incompressible Fluids With Shear, Pressure and Density Dependent Viscosity from Point of View of Continuum Physics 1. Introduction 1.1. What is a fluid? The meaning of words provided in even the most advanced of dictionaries, say the Oxford English Dictionary [1], will rarely serve the needs of a scientist or technologist adequately and this is never more evident than in the case of the meaning assigned to the word “fluid” in its substantive form: “A substance whose particles move freely among themselves, so as to give way before the slightest pressure.” The inadequacy, in the present case, stems from the latter part of the sentence which states that fluids cannot resist pressure; more so as the above definition is immediately followed by the classification: “Fluids are divided into liquids which are incompletely elastic, and gases, which are completely so.”. With regard to the first definition, as “Fluids” obviously include liquids such as water, which under normal ranges of pressure are essentially incompressible and can support a purely spherical state of stress without flowing the definition offered in the dictionary is, if not totally wrong†, at the very least confounding. Much, if not all of hydrostatics † One could take the point of view that no body is perfectly incompressible and thus the body does deform, ever so slightly, due to the application of pressure. The definition however cannot be developed thusly as the intent is clearly that the body suffers significant deformation due to the slightest application of the pressure. $LaTeX: 2005/4/20 $ 8 J. Málek and K. R. Rajagopal is based on the premise that most liquids are incompressible. Next, with regard to the classification of liquids being ”incompletely elastic”, we have to bear in mind that all real gases are not ”completely elastic”. The ideal gas model is of course purely elastic. What then does one mean by a fluid? When we encounter the word “Fluid” for the first time in a physics course at school, we are told that a “fluid” is a body that takes the shape of a container. This meaning assigned to a fluid, can after due care, be used to conclude that a fluid is a body whose symmetry group is the unimodular group†. Such a definition is also not without difficulty. While a liquid takes the shape of the container partially if its volume is less than that of the container, a gas expands to always fill a container. The definition via symmetry groups can handle this difficulty in the sense that it requires densities to be constant while determining the symmetry group. However, this places an unnecessary restriction with regard to defining gases, as this is akin to defining a body on only a small subclass of processes that the body can undergo. We shall not get into a detailed discussion of these subtle issues here. Another definition for a fluid that is quite common, specially with those conversant with the notion of stress, is that a fluid is a body that cannot support a shear stress, as opposed to pressure as required by the definition in [1]. A natural question that immediately arises is that of time scales. How long can a fluid body not support a shear stress? How does one measure this inability to support a shear stress? Is it with the naked eye or is it to be inferred with the aid of sophisticated instruments? Is the assessment to be made in one second, one day, one month or one year? These questions are not being raised merely from the philosophical standpoint. They have very practical pragmatic underpinnings. It is possible, say in the time scale of one hour, one might be unable to discern the flow or deformation that a body undergoes, with the naked eye. This is indeed the case with regard to the experiment on asphalt that has been going on for over seventy years (see Murali Krishnan and Rajagopal [62] for a description of the experiment). The earlier definition for the fluid cannot escape the issue of time scale either. One has to contend with how long it takes to attain the shape of the container. † This statement is not strictly correct. A special subclass of fluids, those that are referred to as “Simple fluids” admit such an interpretation (see Noll [101], Truesdell and Noll [143]). However, it is possible that there exist anisotropic fluids whose symmetry group is not the unimodular group (see Rajagopal and Srinivasa [113]). $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 9 The importance of the notion of time scales was recognized by Maxwell. He observes [89]: “In the case of a viscous fluid it is time which is required, and if enough time is given, the very smallest force will produce a sensible effect, such as would require a very large force if suddenly applied. Thus a block of pitch may be so hard that you cannot make a dent in it by striking it with your knuckles; and yet it will in the course of time flaten itself by its weight, and glide downhill like a stream of water”. The key words in the above remarks of Maxwell are “if enough time is given”. Thus, what we can infer at best is whether a body is more or less fluid-like, i.e., within the time scales of the observation of our interest does a small shear stress produce a sensible deformation or does it not. Let us then accept to “understand” a “Fluid” as a body that, in the time scale of observation of interest, undergoes discernible deformation due to the application of a sufficiently small shear stress‡. 1.2. Navier-Stokes fluid model The popular Navier-Stokes model traces its origin to the seminal work of Newton [99] followed by the penetrating studies by Navier [92], Poisson [103] and StVenant [120], culminating in the definitive study of Stokes [135]†. In his immortal Principia, Newton [99] states: “The resistance arising from the want of lubricity in parts of the fluid is, other things being equal, proportional to the velocity with which the parts of the fluid are separated from one another.” What is now popularly referred to as the Navier-Stokes model implies a linear relationship between the shear stress and the shear rate. However, it was recognized over a century ago that this want of lubricity need not be proportional to the shear stress. Trouton [141] observes “the rate of flow of the material under shearing stress cannot be in simple proportion to shear rate”. However, the popular view persisted and was that ‡ We assume we can agree on what we mean by the time scale of observation of interest. It is also important to recognize that if the shear stress is too small, its effect, the flow, might not be discernible. Thus, we also have to contend with the notion of a spatial scale for a discerning movement and a force scale for discerning forces. † It is interesting to observe what Stokes [135] has to say concerning the development of the fluid model that is referred to as the Navier-Stokes model. Stokes remarks: “I afterward found that Poisson had written a memoir on the same subject, and on referring to it found that he had arrived at the same equations. The method which he employed was however so different from mine that I feel justified in laying the later before this society . . . . The same equations have been obtained by Navier in the case of an incompressible fluid (Mém. de l’Académie, t. VI, p. 389), but his principles differ from mine still more than do Poisson’s.” $LaTeX: 2005/4/20 $ 10 J. Málek and K. R. Rajagopal the rate of flow was proportional to the shear stress as evidenced by the following remarks of Bingham [10]: “ When viscous substance, either a liquid or a gas, is subjected to a shearing stress, a continuous deformation results which is, within certain restrictions directly proportional to the shearing stress. This fundamental law of viscous flow . . . .” Though Bingham offers a caveat “within certain restrictions”, his immediate use of the terms “fundamental law of viscous flow” clearly indicates how well the notion of the proportional relations between a kinematical measure of flow and the shear stress was ingrained in the fluid dynamicist of those times. We will record below, for the sake of discussion, the classical fluid models that bear the names of Euler, and Navier and Stokes. Homogeneous Compressible Euler Fluid: T = −p(%)I . (A.1.1) Homogeneous Incompressible Euler Fluid: T = −pI , trD = 0 . (A.1.2) Homogeneous Compressible Navier-Stokes Fluid: T = −p(%)I + λ(%) (trD) I + 2µ(%)D . (A.1.3) Homogeneous Incompressible Navier-Stokes Fluid: T = −pI + 2µD , trD = 0 . (A.1.4) In the above definitions, T denotes the Cauchy stress, % is the density, λ and µ the bulk and shear moduli of viscosity and D the symmetric part of the velocity gradient. In equations (A.1.1) and (A.1.3) the pressure is defined through an equation of state, while in (A.1.2) and (A.1.4), it is the reaction force due to the constraint that the fluid be incompressible. Within the course of this article we will confine our mathematical discussion mainly to the incompressible Navier-Stokes fluid model (A.1.4) and many of its generalizations. A model that is not of the form (A.1.3) and (A.1.4) falls into the category of (compressible and incompressible) non-Newtonian fluids†. This exclusive definition † Navier-Stokes fluids are usually referred to in the fluid mechanics literature as Newtonian fluids. The equations of motions for Newtonian fluids are referred to as the Navier-Stokes equations. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 11 leads to innumerable fluid models and choices amongst them have to be based on observed response of real fluids that cannot be adequately captured by the above models. This leads us to a discussion of these observations. 1.3. Departures From Newtonian Behavior We briefly list several typical non-Newtonian responses. In their description, detailed characterizations are given to those phenomena and corresponding models whose mathematical properties will be discussed in this paper. A reader interested in a more details on non–Newtonian fluids is referred for example to the monographs Truesdell and Noll [143], Schowalter [125] or Huilgol [54], or to the articles of J.M. Burgers in [14], or to the review article by Rajagopal [115]. Shear-Thinning/Shear-Thickening Let us consider an unsteady simple shear flow in which the velocity field v is given by v = u(y, t)i , (A.1.5) in a Cartesian coordinate system (x, y, z) with base vectors (i, j, k), respectively, t denoting the time. We notice that (A.1.5) automatically meets div v = trD = 0, (A.1.6) and the only non-zero component for the shear stress corresponding to (A.1.3) or (A.1.4) is given by Txy (y, t) = µu,y (y, t) where u,y := du , dy (A.1.7) i.e., the shear stress varies proportionally with respect to the gradient of the velocity, the constant of proportionality being the viscosity. Thus, the graph of the shear stress versus the velocity gradient (in this case the shear rate) is a straight line (see curve 3 in Fig. 1). Let us consider a steady shearing flow, i.e., a flow wherein u = u(y) and κ := u,y = constant at each point of the container occupied by the fluid. It is observed that in many fluids there is a considerable departure from the above relationship (A.1.7) between the shear stress and the shear rate. In some fluids it is observed that the relationship is as depicted by the curve 1 in Fig. 1, i.e. the generalized $LaTeX: 2005/4/20 $ 12 J. Málek and K. R. Rajagopal Figure 1. Shear Thinning/Shear Thickening Figure 2. Generalized viscosity viscosity which is defined through µg (κ) := Txy , κ (A.1.8) is monotonically increasing (cf. curve 1 in Fig. 2). Thus, in such fluids, the viscosity increases with the shear rate and they are referred to as shear-thickening fluids. On the other hand there are fluids for which the relationship between the shear stress and the shear rate is as depicted by curve 2 in Fig. 1. In such fluids, the generalized viscosity decreases with increasing shear rate and for this reason such fluids are called shear-thinning fluids. The Newtonian fluid is thus a very special fluid. It neither shear thins nor shear thickens. The models with shear dependent viscosity are used in many areas of engineering science such as geophysics, glaciology, colloid mechanics, polymer mechanics, blood and food rheology, etc. An illustrative list of references for such models and their applications is given in [87]. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 13 Normal Stress Differences In Simple Shear Flows Next, let us compute the normal stresses along the x, y and z direction for the simple shear flow (A.1.5). A trivial calculation leads to, in the case of models (A.1.3) and (A.1.4), Txx = Tyy = Tzz = −p , and thus Txx − Tyy = Txx − Tzz = Tyy − Tzz = 0 . That is the normal stress differences are zero in a Navier-Stokes fluid. However, it can be shown that some of the phenomena that are observed during the flows of fluids such as die-swell, rod-climbing, secondary flows in cylindrical pipes of noncircular cross-section, etc., have as their basis non-zero differences between these normal stresses. Stress-Relaxation When subject to a step change in strain ε (see Fig. 3 left), that results in a simple shear flow (A.1.5) to ε̇ drawn at Fig. 3 right the stress σ := Txy in bodies modeled by (A.1.3) and (A.1.4) suffers an abrupt change that is undefined at the instant the strain has suffered a change and is zero at all other instants (see Fig. 4 right). On the other hand, there are many bodies that respond in the manner shown in Fig. 5. The graph at right depicts the fluid-like behavior as no stress is necessary to maintain a fixed strain, in the long run. The graph at left represents solid-like response. The Newtonian fluid model is incapable of describing stress-relaxation, a phenomenon exhibited by many real bodies. The important fact to recognize is that a Newtonian fluid stress relaxes instantaneously (see Fig. 4 right)†. Creep Next, let us consider a body that is subject to a step change in the stress (see Fig. 6). In the case of a Newtonian fluid the strain will increase linearly with time (see Fig. 7 at right). However, there are many bodies whose strain will vary as depicted in Fig. 8. The curve at left depicts solid-like behavior while the curve at right depicts fluid-like behavior. The response, which is referred to as “creep” as † This does not mean that it has instantaneous elasticity. $LaTeX: 2005/4/20 $ 14 J. Málek and K. R. Rajagopal Figure 3. Stress-Relaxation test: response to a step change in strain (picture at left). The picture at right sketches its derivative. Figure 4. Shear Stress Response to a step change in strain for linear spring (at left) and Navier-Stokes fluid (at right) Figure 5. Stress-Relaxation for more realistic materials the body flows while the stress is held constant. A Newtonian fluid creeps linearly with time. Many real fluids creep non-linearly with time. Jump discontinuities in stresses Yield stress Bodies that have a threshold value for the stress before they can flow are supposed to exhibit the phenomenon of “yielding”, see Fig. 9. However, if one takes the point of view that a fluid is a body that cannot sustain shear, then by definition there can be no stress threshold to flow, which is the basic premise of the notion of $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 15 Figure 6. Creep test Figure 7. Deformation response to step change of shear stress for linear spring (left) and Newtonian fluid (right) Figure 8. Creep of solid-like and fluid-like materials a “yield stress”. This is yet another example where the importance of time scales comes into play. It might seem, with respect to some time scale of observation, that the flow in a fluid is not discernible until a sufficient large stress is applied. This does not mean that the body in question can support small values of shear stresses, indefinitely. It merely means that the flow that is induced is not significant. A Newtonian fluid has no threshold before it can start flowing. A material responding as a Newtonian fluid once the yield stress is reached is called the Bingham fluid. Activation criterion It is possible that in some fluids, the response characteristics can change when a certain criterion, that could depend on the stress, strain rate or other kinematical quantitites, is met. An interesting example of the same is phenomena of coagulation $LaTeX: 2005/4/20 $ 16 J. Málek and K. R. Rajagopal Figure 9. Yield Stress Figure 10. Activation and Deactivation of Fluids with Shear Dependent Viscosity modelled as jump discontinuities in stress or dissolution of blood. Of course, here issues are more complicated as complex chemical reactions are taking place in the fluid of interest, blood. Platelet activation is followed by their interactions with a variety of plasma proteins that leads to the aggregation of platelets which in turn leads to coagulation, i.e., the formation of clots. The activated platelets also serve as sites for enzyme complexes that play an important role in the formation of clots. These clots, as well as the original blood, are viscoelastic fluids, the clot being significantly more viscous that regular blood. In many situations the viscoelasticity is not consequential and can be ignored and the fluid can be approximated as a generalized Newtonian fluid. While the formation of the clot takes a finite length of time, we can neglect this with respect to a time scale of interest associated with the flowing blood. As the viscosity has increased considerably over a sufficiently short time, in the simple shear flow, the fluid could be regarded as suffering a jump discontinuity as depicted in Fig. 10 at left. On deforming the clot further, we notice a most interesing phenomenon. At a sufficiently high stress, dissolution of the clot takes place and the viscosity now undergoes a significant decrease close to its original value as depicted in Fig. 10 at right . Thus, in general ”activation” can lead to either an increase of decrease in $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 17 viscosity over a very short space of time whereby we can think of it as a jump. See [3] for more details. Pressure-Thickening fluids - Fluids With Pressure Dependent Viscosities Except for the activation criterion, the above departures from Newtonian response are at the heart of what is usually referred to as non-Newtonian fluid mechanics. We now turn to a somewhat different departure from the classical Newtonian model. Notice that the models (A.1.3) and (A.1.4) are explicit expressions for the stress, in terms of kinematical variable D, and the density % in the case of (A.1.3). If the equation of state relating the “thermodynamic pressure” p and the density % is invertible, then we could express λ and µ as functions of the pressure. Thus, in the case of a compressible Navier-Stokes fluid the viscosity µ clearly depends on the pressure. The question to ask is if, in fluids that are usually considered as incompressible liquids such as water under normal operating conditions, the viscosity could be a function of the pressure? The answer to this question is an unequivocal yes by virtue of the fact that when the range of pressures to which the fluid is subject to is sufficiently large, while the density may vary by a few percent, the viscosity could vary by several orders of magnitude, in fact as much a factor of 108 ! Thus, it is reasonable to suppose a liquid to be incompressible while at the same time the viscosity is pressure dependent. In the case of an incompressible fluid whose viscosity depends on both the pressure (mean normal stress) and the symmetric part of the velocity gradient, i.e., when the stress is given by the representation T = −pI + µ(p, D)D . (A.1.9) As p = − 31 trT, it becomes obvious that we have an implicit relationship between T and D, and the constitutive relation is of the form f(T, D) = 0 , (A.1.10) i.e., we have an implicit constitutive equation. It immediately follows from (A.1.10) that ∂f ∂f Ṫ + Ḋ = 0 , ∂T ∂D (A.1.11) [A(T, D)]Ṫ + [B(T, D)]Ḋ = 0 . (A.1.12) which can be expressed as $LaTeX: 2005/4/20 $ 18 J. Málek and K. R. Rajagopal The constitutive relation (A.1.12) is more general that (A.1.10) as an implicit equation of the form (A.1.12) need not be integrable to yield an equation of the form (A.1.10). A further generalization within the context of implicit constitutive relations for compressible bodies is the equation g(%, T, D) = 0 . (A.1.13) Before we get into a more detailed discussion of implicit models for fluids let us consider a brief history of fluids with pressure dependent viscosity. Stokes [135] recognized that in general the viscosity could depend upon the pressure. It is clear from his discussion that he is considering liquids such as water. Having recognized the dependence of the viscosity on the pressure, he makes the simplifying assumption “If we suppose µ to be independent of the pressure also, and substitute . . . “. Having made the assumption that the viscosity is independent of the pressure, he feels the need to substantiate that such is indeed the case for a restricted class of flows, those in pipes and channels, according to the experiments of DuBuat [26]: “Let us now consider in what cases it is allowable to suppose µ to be independent of the pressure. It has been concluded by DuBuat from his experiments on the motion of water in pipes and canals, that the total retardation of the velocity due to friction is not increased by increasing the pressure . . . . I shall therefore suppose that for water, and by analogy for other incompressible fluids, µ is independent of the pressure.” While the range of pressures attained in DuBuat’s experiment might justify the assumption made by Stokes for a certain class of problems, one cannot in general make such an assumption. There are many technologically significant problems such as elastohydrodynamics (see Szeri [136]) wherein the fluid is subject to such a range of pressure that the viscosity changes by several orders of magnitude. There is a considerable amount of literature concerning the variation of viscosity with pressure and an exhaustive discussion of the literature before 1931 can be found in the authoritative treatise on the physics of high pressure by Bridgman [12]. Andrade [5] suggested the viscosity depends on the pressure, density and temperature in the following manner µ(p, %, θ) = A% 1/2 exp B (p + D%2 ) θ , (A.1.14) $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 19 where A, B and D are constants. In the processes where the temperature is uniformly constant, in the case of many liquids, it would be reasonable to assume that the liquid is incompressible and the viscosity varies exponentially with the pressure. This is precisely the assumption that is made in studies in elastohydrodynamics. One can carry out a formal analysis based on standard representation theorems for isotropic functions (see Spencer [133]) that requires that the (A.1.10) satisfying for all orthogonal tensors Q g(%, QTQT , QDQT ) = Qg(%, T, D)QT take the implicit constitutive relation α0 I + α1 T + α2 D + α3 T2 + α4 D2 + α5 (TD + DT) + α6 (T2 D + DT2 ) + α7 (TD2 + D2 T) + α8 (T2 D2 + D2 T2 ) = 0 , (A.1.15) where the material moduli αi i = 0, . . . , 8 depend on %, trT, trD, trT2 , trD2 , trT3 , trD3 , tr(TD), tr(T2 D), tr(D2 T), tr(T2 D2 ) . The model T = −p(%)I + β(%, trT, trD2 )D is a special subclass of models of the form (A.1.15). The counterpart in the case of an incompressible fluid would be T = −pI + µ(p, trD2 )D , trD = 0. (A.1.16) We shall later provide a thermodynamic basis for the development of the model (A.1.16). 2. Balance equations 2.1. Kinematics We shall keep our discussion of kinematics to a bare minimum. Let B denote the abstract body and let κ : B → E, where E is a three dimensional Euclidean space, be a placer and κ(B) the configuration of the body. We shall assume that the placer is one to one. By a motion we mean a one parameter family of placers (see Noll [100]). It follows that if κR (B) is some reference configuration, and κt (B) a configuration $LaTeX: 2005/4/20 $ 20 J. Málek and K. R. Rajagopal at time t, then we can identify the motion with a mapping χκR : κR (B)×R → κt (B) such that† x = χκR (X, t) . (A.2.1) We shall suppose that χκR is sufficiently smooth to render the operations defined on it meaningful. Since χκR is one to one, we can define its inverse so that X = χ−1 κR (x, t) . (A.2.2) Thus, any (scalar) property ϕ associated with an abstract body B can be expressed as (analogously we proceed for vectors or tensors) ϕ = ϕ(P, t) = ϕ̂(X, t) = ϕ̃(x, t) . (A.2.3) We define the following Lagrangean and Eulerian temporal and spatial derivatives: ϕ̇ := ∂ ϕ̂ , ∂t ϕ,t := ∂ ϕ̃ , ∂t ∂ ϕ̂ , ∂X ∇X ϕ = ∇x ϕ := ∂ ϕ̃ . ∂x (A.2.4) The Lagrangean and Eulerian divergence operators will be expressed as Div and div, respectively. The velocity v and the acceleration a are defined through v= ∂χκR ∂t a= ∂ 2 χ κR , ∂t2 (A.2.5) and the deformation gradient FκR is defined through F κR = ∂χκR . ∂X (A.2.6) The velocity gradient L and its symmetric part D are defined through L = ∇x v , D= 1 (L + LT ) . 2 (A.2.7) It immediately follows that L = ḞκR F−1 κR . (A.2.8) It also follows from the notations and definitions given above, in particular from (A.2.4) and (A.2.5) that ϕ̇ = ϕ,t + ∇x ϕ · v . (A.2.9) † It is customary to denote x and X which are points in an Euclidean space as bold face quantities. We however choose not to do so. On the other hand, all vectors, and higher order tensors are indicated by bold face. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 21 2.2. Balance of Mass - Incompressibility - Inhomogenity Z The balance of mass in its Lagrangean form states that Z %R (X)dX = %(x, t)dx for all PR ⊂ κR (B) with Pt := χκR (PR , t) , PR Pt (A.2.10) which immediately leads to, using the Substitution theorem, %(x, t) det FκR (X, t) = %R (X) . A body is incompressible if Z Z dX = dx for all PR ⊂ κR (B) det FκR (X, t) = 1 for all X ∈ κR (B) . PR (A.2.11) Pt which leads to (A.2.12) If det FκR is continuously differentiable with respect to time, then by virtue of the identity we conclude, since det FκR d det FκR = div v det FκR , dt 6= 0 that div v(x, t) = 0 for all t ∈ R and x ∈ κt (B) . (A.2.13) It is usually in the above form that the constraint of incompressibility is enforced in fluid mechanics. From the Eulerian perspective, the balance of mass takes the form Z d % dx = 0 for all Pt ⊂ κt (B) . dt Pt (A.2.14) It immediately follows that %,t + (∇x %) · v + % div v = 0 ⇐⇒ %,t + div(%v) = 0 . (A.2.15) If the fluid is incompressible, it immediately follows from (A.2.15) that %,t + (∇x %) · v = 0 ⇐⇒ %̇ = 0 ⇐⇒ %(t, x) = %(0, X) = %R (X) . (A.2.16) That is, for a fixed particle, the density is constant, as a function of time. However, the density of a particle may vary from one particle to another. The fact that the density varies at certain location in space, does not imply that the fluid is not incompressible. This variation is due to the fact that the fluid is inhomogeneous, a concept that has not been grasped clearly in fluid mechanics (see Anand and Rajagopal [4] for a discussion). $LaTeX: 2005/4/20 $ 22 J. Málek and K. R. Rajagopal 2.3. Balance of Linear Momentum The balance of linear momentum that originates from the second law of Newton in classical mechanics applied to each subset Pt = χκR (PR , t) of the current configuration takes the form Z Z Z d ρv dx = ρb dx + TT n dS , dt Pt Pt ∂Pt (A.2.17) where T denotes the Cauchy stress that is related to the surface traction t through t = TT n, and b denotes the specific body force. It then leads to the balance of linear momentum in its local Eulerian form: %v̇ = div TT + %b . (A.2.18) Two comments are in order. First, considering the case when κt (B) = κR (B) for all t ≥ 0 and setting Ω := κR (B), it is not difficult to conclude at least for incompressible fluids, that (A.2.17) and (A.2.14) imply that Z Z h Z i d T ρv dx + (ρv)(v · n) − T n dS = ρb dx , dt O ∂O O Z Z d ρ(v · n)dS = 0, ρ dx + dt O ∂O (A.2.19) (A.2.20) valid for all (fixed) subsets O of Ω. When compared to (A.2.17), this formulation is more suitable for further consideration in those problems where the velocity field v is taken as a primitive field defined on Ω × h0, ∞) (i.e. it is not defined through (A.2.5)). To illustrate this convenience, we give a simple analogy from classical mechanics: consider a motion of a mass-spring system described by the second order ordinary differential equations for the displacement from the equilibrium and compare it with a free fall of the mass captured by the first order ordinary differential equations for the velocity. In fluid mechanics, the velocity field is typically taken as primitive variable. Second, the derivation of (A.2.18) from (A.2.17) and similarly (A.2.15) from (A.2.14) requires certain smoothness of particular terms. In analysis, the classical formulations of the balance equations (A.2.18) and (A.2.15) are usually starting points for definition of various kinds of solutions. Following Oseen [102] (see also [34], [35]), we want to emphasize that the notion of a weak solution (or suitable $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 23 weak solution) is very natural for equations of continuum mechanics, since their weak formulation can be directly obtained from the original formulations of the balance laws (A.2.14) and (A.2.17) or better (A.2.19) and (A.2.20). This comment is equally applicable to the other balance equation of continuum physics as well. 2.4. Balance of Angular Momentum In the absence of internal couples, the balance of angular momentum implies that the Cauchy stress is symmetric, i.e., T = TT . (A.2.21) 2.5. Balance of Energy The local form of the balance of energy is %˙ = T · ∇v − div q + %r , (A.2.22) where denotes the internal energy, q denotes the heat flux vector and r the specific radiant heating. 2.6. Further Thermodynamic Considerations (The Second Law). Reduced dissipation equation To know how a body is constituted and to distinguish one body from another, we need to know how bodies store energy. How, and how much of, this energy that is stored in a body can be recovered from the body. How much of the working on a body is converted to energy in thermal form (heat). What is the nature of the latent energy that is associated with the changes in phase that the body undergoes. What is the nature of the latent energy (which is different in general from latent heat). By what different means does a body produce the entropy? These are but few of the pieces of information that one needs to describe the response of the body. Merely knowing this information is insufficient to describe how the body will respond to external stimuli. A body’s response has to meet the basic balance laws of mass, linear and angular momentum, energy and the second law of thermodynamics. Various forms for the second law of thermodynamics have been proposed and are associated with the names of Kelvin, Plank, Clausius, Duhem, Caratheodory and others. Essentially, the second law states that the rate of entropy production has to $LaTeX: 2005/4/20 $ 24 J. Málek and K. R. Rajagopal be non-negative†. A special form of the second law, the Claussius-Duhem inequality, has been used, within the context of a continua, to obtain restrictions on allowable constitutive relations (see Coleman and Noll [20]). This is enforced by allowing the body to undergo arbitrary processes in which the second law is required to hold. The problem with such an approach is that the constitutive structure that we ascribe to a body is only meant to hold for a certain class of processes. The body might behave quite differently outside this class of processes. For instance, while rubber may behave like an elastic material in the sense that the stored energy depends only on the deformation gradient and this energy can be completely recovered in processes that are reasonably slow in some sense, the same rubber if deformed at exceedingly high strain rates crystallizes and not only does the energy that is stored not depend purely on the deformation gradient, all the energy that was supplied to the body cannot be recovered. Thus, the models for rubber depend on the process class one has in mind and this would not allow one to subject the body to arbitrary process. We thus find it more reasonable to assume the constitutive structures for the rate of entropy production, based on physical grounds, that are automatically non-negative. Let us first introduce the second law of thermodynamics in the form %θη̇ ≥ − div q + q · (∇x θ) + %r , θ (A.2.23) where η denotes the specific entropy. On introducing the specific Helmholtz potential ψ through ψ := − θη , and using the balance of energy (A.2.22), we can express (A.2.23) as T · L − %ψ̇ − %θ̇η − q · (∇x θ) ≥ 0. θ (A.2.24) The above inequality is usually referred to as the dissipation inequality. This inequality is commonly used in continuum mechanics to obtain restrictions on the constitutive relations. A serious difficulty with regard to such an approach becomes immediately apparent. No restrictions whatsoever can be placed on the radiant heating. More importantly, the radiant heating is treated as a quantity that adjusts itself to meet the balance of energy. But this is clearly unacceptable as the † There is a disagreement as to whether this inequality ought to be enforced locally at every point in the body, or only globally, even from the point of view of statistical thermodynamics. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 25 radiant heating has to be a constitutive specification. How a body responds to radiant heating is critical, especially in view of the fact that all the energy that our world receives is in the form of electromagnetic radiation which is converted to energy in its thermal form (see Rajagopal and Tao [114] for a discussion of these issues). As we shall be primarily interested in the mechanical response of fluids, we shall ignore the radiant heating altogether, but we should bear in mind the above observation when we consider more general processes. We shall define the specific rate of entropy production ξ through ξ := T · L − %ψ̇ − %θ̇η − q · (∇x θ) . θ (A.2.25) We shall make constitutive assumptions for the rate of entropy production ξ and require that (A.2.25) hold in all admissible processes (see Green and Nagdhi [48]). Thus, the equation (A.2.25) will be used as a constraint that is to be met in all admissible processes. We shall choose ξ so that it is non-negative and thus the second law is automatically met. We now come to a crucial step in our thermodynamic considerations. From amongst a class of admissible non-negative rate of entropy productions, we choose that which is maximal. This is asking a great deal more than the second law of thermodynamics. The rationale for the same is the following. Let us consider an isolated system. For such a system, it is well accepted that its entropy becomes a maximum and the system would reach equilibrium. The assumption that the rate of entropy production is a maximum ensures that the body attains its equilibrium as quickly as possible. Thus, this assumption can be viewed as an assumption of economy or an assumption of laziness, the system tries to get to the equilibrium state as quickly as possible, i.e., in the most economic manner. It is important to recognize that this is merely an assumption and not some deep principle of physics. The efficacy of the assumption has to be borne out by its predictions and to date the assumption has led to meaningful results in a wide variety of material behavior (see results pertinent to viscoelasticity [112], [113], classical plasticity ([109], [110]), twinning ([107], [108]), solid to solid phase transition [111]), crystallization in polymers ([117], [118]), single crystal supper alloys [104], etc.). $LaTeX: 2005/4/20 $ 26 J. Málek and K. R. Rajagopal 2.7. Isothermal flows at uniform temperature Here, we shall restrict ourselves to flows that take place at constant temperature for the whole period of interest at all points of the body. Consequently, the equations governing such flows for a compressible fluid are %̇ = −% div v %v̇ = div T + %b , (A.2.26) while for an incompressible fluid they take the form div v = 0 , %̇ = 0 , %v̇ = div T + %b . (A.2.27) Note that (A.2.24) and (A.2.25) reduce to T · D − %ψ̇ = ξ and ξ ≥ 0 . (A.2.28) In order to obtain a feel for the structure of the constitutive quantities appearing in (A.2.28), we consider first the Cauchy stress for the incompressible and compressible Euler fluid, and then for the incompressible and compressible NavierStokes fluid. Note that Euler fluids are ideal fluids in that there is no dissipation in any process undergone by the fluid, i.e., ξ ≡ 0 in all processes. Compressible Euler fluid. Since ξ ≡ 0 and (A.1.1) implies T · L = −p(%)I · L = −p(%)trL = −p(%)trD = −p(%) div v , the reduced thermo-mechanical equation (A.2.28) simplifies to %ψ̇ = −p(%) div v . (A.2.29) This suggests that it might be appropriate to consider ψ of the form ψ = Ψ(%) . (A.2.30) In fact, since an ideal fluid is an elastic fluid, it follows that its specific Helmoltz free energy ψ depends only on the deformation gradient F. If we suppose that the symmetry group of a fluid is the unimodular group, then the balance of mass could lead one to conclude that ψ depends on the density %. Using (A.2.26)1 , we then have from (A.2.30) ψ̇ = Ψ,% (%)%̇ = −%Ψ,% (%) div v , (A.2.31) $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 27 and we conclude from (A.2.29) and (A.2.31) that p(%) = %2 Ψ,% (%) . (A.2.32) Incompressible Euler fluid. Since we are dealing with a homogeneous fluid we have % ≡ %∗ , where %∗ is a positive constant. We also have Ψ̇(%∗ ) = 0 , T · L = −pI · L = −p(%) div v = 0 , and ξ ≡ 0 . Thus, each term in (A.2.28) vanishes and (A.2.28) clearly holds. Compressible Navier-Stokes fluid. Consider T of the form (A.1.3) and ψ of the form (A.2.30) fulfilling (A.2.32). Denoting Cδ the deviatoric (traceless) part of any tensor C, i.e., Cδ = C − 31 (trC)I, then we have ξ = T · L − %ψ̇ = −p(%) div v + 2µ(%)D · D + λ(%)(trD)2 + %2 Ψ,% (%) = 2µ(%)D · D + λ(%)(trD)2 2 = 2µ(%)Dδ · Dδ + λ(%) + µ(%) (trD)2 . 3 2 Note that the second law of thermodynamics is met if µ(%) ≥ 0 and λ(%) + µ(%) ≥ 3 0. Incompressible Navier-Stokes fluid. Similar considerations as those for the case of a compressible Navier-Stokes fluid imply ξ = 2µD · D = 2µ|D|2 . Note that for both the incompressible Euler and Navier-Stokes fluid we have 1 p = − trT . 3 2.8. Natural Configurations Most bodies can exist stress free in more than one configuration and such configurations are referred to as ”natural configurations” (see Eckart [28], Rajagopal [116]).Given a current configuration of a homogeneously deformed body, the stress-free configuration that the body takes on upon the removal of all external stimuli is the underlying ”natural configuration” corresponding to the current configuration of the body. As a body undergoes a thermodynamic process, in general, the underlying natural configuration evolves. The evolution of this underlying $LaTeX: 2005/4/20 $ 28 J. Málek and K. R. Rajagopal natural configuration is determined by the maximization of the entropy production (see how this methodology is used in viscoelasticity ([112], [113]), classical plasticity ([109], [110]), twinning ([107], [108]), solid to solid phase transition [111]), crystallization in polymers ([117], [118]), single crystal super alloys [104]). In the case of the both incompressible and compressible Navier-Stokes fluids and the generalizations discussed here, the current configuration κt (B) itself serves as the natural configuration. 3. The Constitutive Models For Compressible and Incompressible Navier-Stokes Fluids and Some of their Generalizations 3.1. Standard approach in continuum physics The starting point for the development of the model for a homogeneous compressible Navier-Stokes fluid is the assumption that the Cauchy stress depends on the density and the velocity gradient, i.e., T = f(%, L) . (A.3.1) It follows from the assumption of frame-indifference that the stress can depend on the velocity gradient only through its symmetric part, i.e., T = f(%, D) . (A.3.2) The requirement the fluid be isotropic then implies that f(%, D) = α1 I + α2 D + α3 D2 , (A.3.3) where αi = αi (%, ID , IID , IIID ), and ID = trD, IID = 1 [(trD)2 − trD2 ] , 2 IIID = det D . If we require that the stress be linear in D, then we immediately obtain T = −p(%)I + λ(%)(trD)I + 2µ(%)D , (A.3.4) which is the classical homogeneous compressible Navier-Stokes fluid. Starting with the assumption that the fluid is incompressible and homogeneous, and T = g(L) (A.3.5) $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 29 a similar procedure leads to (see Truesdell and Noll [143]) T = −pI + 2µD . (A.3.6) The standard procedure for dealing with constraints such as incompressibility, namely that the constraint reactions do no work (see Truesdell [142]) is fraught with several tacit assumptions (we shall not discuss them here) that restrict the class of models possible. For instance it will not allow the material modulus µ to depend on the Lagrange multiplier p. The alternate approach presented below attempts to avoid such drawbacks. Another general alternative procedure established in purely mechanical context has been recently developed in Rajagopal and Srinivasa [106]. 3.2. Alternate approach We provide below an alternate approach for deriving the constitutive relation for a homogeneous compressible and an incompressible Navier-Stokes fluid. Instead of assuming a constitutive equation for the stress as the starting point, we shall start assuming forms for the Helmholtz potential and the rate of dissipation, namely two scalars. We first focus on the derivation of the constitutive equation for the Cauchy stress for the compressible Navier-Stokes fluid supposing that ψ(x, t) = Ψ(%(x, t)) . (A.3.7) and ξ = Ξ(D) = 2 µ(%)D · D + λ(%)(trD)2 2 = 2 µ(%)|Dδ |2 + (λ(%) + µ(%))(trD)2 , 3 2 where µ(%) ≥ 0 , λ(%) + µ(%) ≥ 0 . 3 (A.3.8) With such a choice of ξ the second law is automatically met, and (A.2.28) takes the form (cf. (A.2.31)) ξ = (T + %2 Ψ,% (%)I) · D . (A.3.9) For a fixed T there are plenty of D’s that satisfy (A.3.8) and (A.3.9). We pick a D such that D maximizes (A.3.8) and fulfils (A.3.9). This leads to a constrained maximization that gives the following necessary condition ∂Ξ ∂Ξ − λ1 (T + %2 Ψ,% (%)I − ) = 0, ∂D ∂D $LaTeX: 2005/4/20 $ 30 J. Málek and K. R. Rajagopal or equivalently 1 + λ1 ∂Ξ = (T + %2 Ψ,% (%)I) . (A.3.10) λ1 ∂D To eliminate the constraint we take scalar product of (A.3.10) with D. Using (A.3.9), (A.3.10) and the fact that ∂Ξ = 2(2µ(%)D + λ(%)(trD)I) , ∂D we find that 1 + λ1 = λ1 1 Ξ = . 2 ·D ∂Ξ ∂D (A.3.11) (A.3.12) Inserting (A.3.11) and (A.3.12) into (A.3.10) we obtain T = −%2 Ψ,% (%)I + 2µ(%)D + λ(%)(trD)I . (A.3.13) Finally, setting p(%) = %2 Ψ,% (%) we obtain the Cauchy stress for compressible Navier-Stokes fluid, cf. (A.1.3). Next, we provide a derivation for an hierarchy of incompressible fluid models that generalize the incompressible Navier-Stokes fluid in the following sense: the viscosity can not only be a constant, but it can be a function that may depend on the density, the symmetric part of the velocity gradient D specifically through D · D, or the mean normal stress, i.e. the pressure p := − 13 trT, or it can depend on any or all of them. We shall consider the most general case within this setting by assuming that ξ = Ξ(p, %, D) = 2ν(p, %, D · D) D · D . (A.3.14) Clearly, if ν ≥ 0 then automatically ξ ≥ 0, ensuring that the second law is complied with. We assume that the specific Helmoltz potential ψ is of the form (A.3.7). By virtue of the fact that the fluid is incompressible, i.e., trD = 0 , (A.3.15) we obtain %̇ = 0, ψ̇ vanishes in (A.2.28) and we have from (A.2.28) T · D = Ξ. (A.3.16) Following the same procedure as that presented above, in case of a compressible fluid, we maximize Ξ with respect to D that is subject to the constraints (A.3.15) and (A.3.16). As the necessary condition for the extremum we obtain the equation (1 + λ1 )Ξ,D − λ1 T − λ0 I = 0 , (A.3.17) $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 31 where λ0 and λ1 are the Lagrange multipliers due to the constraints (A.3.15) and (A.3.16). We eliminate them as follows. Taking the scalar product of (A.3.17) with D, and using (A.3.15) and (A.3.16) we obtain 1 + λ1 Ξ = . λ1 Ξ,D · D (A.3.18) Note that Ξ,D = 4 ν(p, %, D · D) + ν,D (p, %, D · D)D · D D . (A.3.19) Consequently, trΞ,D = 0 by virtue of (A.3.15). Thus, taking the trace of (A.3.17) we have − λ0 = −p λ1 1 with p = − trT . 3 (A.3.20) Using (A.3.17)–(A.3.20), we finally find that (A.3.17) takes the form T = −pI + 2 ν(p, %, D · D)D . (A.3.21) Mathematical issues related to the system (A.2.27) with the constitutive equation (A.3.21) will be discussed in the second part of this treatise. The fluid given by (A.3.21) has the ability to shear thin, shear thicken and pressure thicken. After adding the yield stress or activation criterion, the model could capture phenomena connected with the development of discontinuous stresses. On the other hand, the model (A.2.27) together with (A.3.21) cannot stress relax or creep in a non-linear way, neither can it exhibit nonzero normal stress differences in a simple shear flow. 4. Boundary Conditions No aspect of mathematical modeling has been neglected as that of determining appropriate boundary conditions. Mathematicians seem especially oblivious to the fact that boundary conditions are constitutive specifications. In fact, boundary conditions require an understanding of the nature of the bodies that are divided by the boundary. Boundaries are rarely sharp, with the constituents that abut either side of the boundary invariably exchanging molecules. In the case of the boundary between two liquids or a gas and a liquid this molecular exchange is quite obvious, it is not so in the case of a reasonably impervious solid boundary and a liquid. The ever popular “no-slip” (adherence) boundary condition is supposed to have had the imprimatur of Stokes behind it, but Stokes’ opinions concerning the status of the $LaTeX: 2005/4/20 $ 32 J. Málek and K. R. Rajagopal “no-slip” condition are nowhere close to unequivocal as many investigators lead one to believe. A variety of suggestions were put forward by the pioneers of the field, Bernoulli, DuBuat, Navier, Poisson, Girard, Stokes and others, as to the condition that ought to be applied on the boundary between an impervious solid and a liquid. One fact that was obvious to all of them was that boundary conditions ought to be derived, just as constitutive relations are developed for the material in the bulk, even more so. This is made evident by Stokes [135] who makes this obvious with his remarks: “Besides the equations which must hold good at any point in the interior of the mass, it will be necessary to form also the equations which must be satisfied at the boundary.” After emphasizing the need to derive the equations that ought to be applied at a boundary, Stokes [135] goes on to derive a variety of such boundary conditions. That Stokes [135] was in two minds about the appropriateness of the “no-slip” boundary condition is evident from his following remarks: “DuBuat found by experiment that when the mean velocity of water flowing through a pipe is less than one inch in a second, the water near the inner surface of the pipe is at rest. If these experiments may be trusted, the conditions to be satisfied in the case of small velocities are those which first occurred to me . . . .”, but he goes on to add: “I have said that when the velocity is not small the tangential force called into action by the sliding of water over the inner surface of the pipe varies nearly the square of the velocity . . . .”. The key words that demand our attention are “the sliding of water over the inner surface”. Sliding implies that Stokes believed that the fluid is slipping at the boundary. That he was far from convinced concerning the applicability of the “no-slip” condition is made crystal clear when he remarks: “The most interesting questions concerning the subject require for their solution a knowledge of the conditions which must be satisfied at the surface of solid in contact with the fluid, which, except in the case of very small motions, are unknown.”. To Stokes the determination of appropriate boundary conditions was an open problem. An excellent concise history concerning boundary conditions for fluids can be found in Goldstein [47]. We discuss briefly some of the boundary conditions that have been proposed for a fluid flowing past a solid impervious boundary. Navier [92] derived a slip condition which can be duly generalized to the condition v · τ = −K(Tn · τ ) , K ≥ 0, (A.4.1) $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 33 where n is the unit outward normal vector and τ stands for a tangent vector at the boundary point; K is usually assumed to be a constant but it could however be assumed to be a function of the normal stresses and the shear rate, i.e., K = K(Tn · n, |D|2 ) . (A.4.2) The above boundary conditions, when K > 0, is referred to as the slip boundary condition. If K = 0, we obtain the classical “no-slip” boundary condition. Another boundary condition that is sometimes used, especially when dealing with non-Newtonian fluids, is the “threshold-slip” condition. This takes the form |Tn · τ | ≤ α|Tn · n| =⇒ v · τ = 0 , |Tn · τ | > α|Tn · n| =⇒ v · τ 6= 0 and − γ v·n = Tn · τ , |v · n| (A.4.3) where γ = γ(Tn · n, v · τ ). The above condition implies that fluid will not slip until the ratio of the magnitude of shear stress and the magnitude of the normal stress exceeds a certain value. When it does exceed that value, it will slip and the slip velocity will depend on both the shear and normal stresses. It is also possible to require that γ depends on |D|2 . A much simpler condition that is commonly used is v·τ = v0τ 0 if |Tn · τ | > β , (A.4.4) if |Tn · τ | ≤ β . Thus the fluid will slip if the shear stress exceeds a certain value. Here, v0τ are given scalar functions for each τ generating the tangent space at the boundary. If the boundary is permeable, then in addition to the possibility of v · τ not being equal to zero, we have to specify the normal component of the velocity v · n. Several flows have been proposed for flows past porous media, however we shall not discuss them here. In order to understand characteristic features of particular terms appearing in the system of PDEs, as well as their natural dependence, it is convenient to eliminate completely the presence of the boundary and boundary conditions on the flow, i.e., on the solution. This can be realized in two way: $LaTeX: 2005/4/20 $ 34 J. Málek and K. R. Rajagopal 1) Assume that the fluid occupies the whole three-dimensional space with the velocity vanishing at |x| → +∞. Then starting with an initial-condition v(0, ·) = v0 ∈ R3 (A.4.5) we are interested in knowing the properties of the velocity and the pressure of governing equations at any instant of the time t > 0 and any position x ∈ R3 . 2) Assume that for T, L ∈ (0, ∞) vi , p :[0, T ] × R3 → R are L-periodic at each direction xi , Z Z p dx = 0. i = 1, 2, 3 vi dx = 0, with Ω (A.4.6) Ω Here Ω = (0, L) × (0, L) × (0, L) is a periodic cell. The advantage of the second case consists in the fact that we work on domain with a compact closure. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 35 Chapter B Mathematical Analysis of Flows of Fluids With Shear, Pressure and Density Dependent Viscosity 1. Introduction 1.1. A taxonomy of models The objective of this chapter is to provide a survey of results regarding the mathematical analysis of the system of partial differential equations for the (unknown) density ρ, the velocity v = (v1 , v2 , v3 ) and the pressure (mean normal stress) p, partial differential equations being ρ,t + ∇ρ · v = 0, div v = 0, ρ(v,t + div(v ⊗ v)) = −∇p + div(2ν(p, ρ, |D(v)|2 )D(v) + ρb, (B.1.1) focusing however mostly on some of its simplifications specified below. The system (B.1.1) is exactly the system (A.2.27) with the constitutive equations (A.3.21) whose interpretation from the perspective of non-Newtonian fluid mechanics and the connection to compressible fluid models were discussed in Chapter A. Unlike (A.2.27) and (A.3.21) we use a different notation in order to express the equations in the form (B.1.1). First of all, owing to the incompressibility constraint, we have v̇ = v,t + [∇x v]v = v,t + div(v ⊗ v), where the tensor product a ⊗ b is the second order tensor with components (a ⊗ b)ij = ai bj for any a = (a1 , a2 , a3 ), b = (b1 , b2 , b3 ). Next note that in virtue of to (B.1.1)2 , we can rewrite (B.1.1)1 as ρ,t + div(ρv) = 0. We also explicitly use the notation D(v) instead of D in order to clearly identify our $LaTeX: 2005/4/20 $ 36 J. Málek and K. R. Rajagopal interest concerning the velocity field. As discussed in Chapter A, the model (B.1.1) includes a lot of special important cases particularly for homogeneous fluids†. Note that for the case of a homogeneous fluid,(B.1.1) reduces to div v = 0, v,t + div(v ⊗ v) = −∇p + div(2ν(p, |D(v)|2 ))D(v)) + b, (B.1.2) p 1 ρ0 , and relabelling the dynamic pressure ρ0 2 ν(p,ρ0 ,|D(v)| ) again as p and ν(p, |D(v)|2 ), respectively. ρ0 obtained by multiplying (B.1.1)2 by and the dynamic viscosity For later reference, we give a list of several special models contained as a special subclasses of (B.1.2): a) Fluids with pressure dependent viscosity where ν is independent of the shear rate, but depends on the pressure p: div v = 0, v,t + div(v ⊗ v) − div(ν(p)[∇v + (∇v)T ]) = −∇p + b; (B.1.3) b) Fluids with shear dependent viscosity with the viscosity independent of the pressure: div v = 0, v,t + div(v ⊗ v) − div S(D(v)) = −∇p + b; (B.1.4) here we introduce the notation S(D(v)) = 2ν(|D(v)|2 )D(v). (B.1.5) This class of fluids includes: c) Ladyzhenskaya’s fluids‡ with ν(|D(v)|2 ) = ν0 + ν1 |D(v)|r−2 , where r > 2 is fixed, ν0 and ν1 are positive numbers: div v = 0, v,t +div(v⊗v)−ν0 4v−2ν1 div(|D(v)|r−2 D(v)) = −∇p+b (B.1.6) d) Power-law fluids with ν(|D(v)|2 ) = ν1 |D(v)|r−2 where r ∈ (1, ∞) is fixed and ν1 is a positive number: div v = 0, v,t + div(v ⊗ v) − 2ν1 div(|D(v)|r−2 D(v)) = −∇p + b (B.1.7) † Recall that in our setting a fluid is homogeneous if for some positive number ρ 0 the density fulfils ρ(x, t) = ρ0 for all time instants t ≥ 0 and all x ∈ κt (B). ‡ For r = 3 this system of PDEs is frequently called Smagorinski’s model of turbulence, see [131]. Then ν0 is molecular viscosity and ν1 is the turbulent viscosity. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 37 e) Navier-Stokes fluids with ν(p, |D(v)|2 ) = ν0 (ν0 being a positive number): div v = 0, v,t + div(v ⊗ v) − ν0 4v = −∇p + b. (B.1.8) The equations of motions (B.1.8) for a Navier-Stokes fluid are referred to as the Navier-Stokes equations (NSEs), the equations (B.1.6) for a Ladyzhenskaya’s fluid will be referred to as the Ladyzhenskaya’s equations. A fluid captured by Ladyzhenskaya’s equations reduces to NSEs (B.1.8) by taking ν1 = 0 in (B.1.6) and to powerlaw fluids by setting ν0 = 0. Note also that setting r = 2 in the Ladyzhenskaya’s equations we again obtain NSEs with the constant viscosity 2(ν0 + ν1 ). 1.2. Mathematical self-consistency of the models Irrespective of how accurately a model of our choice approximates the real behavior of a fluid, mathematical analysis is interested in questions concerning its mathematical self-consistency‡. We say that a model is mathematically self-consistent if it exhibits at least the following properties: (I) large-time and large-data existence: Completing the model by a reasonable set of boundary conditions and considering a smooth, but arbitrary initial data, the model should admit a solution for all positive time instants. (II) large-time and large-data uniqueness: The motion is fully determined by its initial, boundary and other data and depends on them continuously; particularly, such a motion is unique for a given set of data. (III) large-time and large-data regularity: The physical quantities, such as the velocity in the case of fluids, are bounded. These three requirements thus form a minimal set of mathematical properties that one would like any evolutionary model of (classical) mechanics to exhibit, particularly any of the models (B.1.3) up to (B.1.8). A discussion of the current state of results with regard the properties (I), (II) and (III) for the above models forms the backbone of the remaining part of this article. Towards purpose, we eliminate the influence of the boundary by considering spatially periodic problem only, cf.(A.4.6). On the other hand, we do not apply tools that are just suitable for periodic functions (such as Fourier series) but rather use ‡ See a video record of Caffarelli’s presentation of the 3rd millenium problem ”Navier-Stokes and smoothness”[15]. $LaTeX: 2005/4/20 $ 38 J. Málek and K. R. Rajagopal tools and approaches that can be used under more general conditions for other boundary-value problem, as well. 1.3. Weak solution: a natural notion of solution for PDEs of the continuum physics The tasks (I)-(III) require to know what is meant by solution. We obtain a hint from the balance of linear momentum for each (measurable) subset of the body (A.2.17), as recognized already by Oseen [102]. Note that (A.2.19) requires some integrability of the first derivatives of the velocity and the integrability of the pressure, while the classical formulation† (B.1.8) is based on the knowledge of the second derivatives of v and the gradient of p. Oseen [102] not only observed this discrepancy between (A.2.19) and (B.1.8), but he also proposed and derived a notion of weak solution directly from the original formulation‡ of the balance of linear momentum (A.2.19). To be more specific, following the procedure outlined by Oseen [102] (for other approaches see also [34] p. 55, and [35]) it is possible to conclude directly from (A.2.19) and (A.2.20) that ρ, v and T fulfil for all t > 0 − Z tZ 0 Z (ρv)(τ, x) · ϕ,τ (τ, x) dx dτ + (ρv)(t, x) · ϕ(t, x) dx Ω Z Z tZ (ρv ⊗ v) · ∇ϕ dx dτ − (ρv)(0, x) · ϕ(0, x) dx − 0 Ω Ω Z tZ Z tZ + T · ∇ϕ dx dτ = ρb · ϕ dx dτ Ω 0 ∞ for all ϕ ∈ D −∞, +∞; Cper − Z tZ 0 Ω 3 0 Ω and Z ρ(τ, x)ξ,t (τ, x) dx dτ + ρ(t, x)ξ(t, x) dx Ω Ω Z Z tZ − ρ(0, x)ξ(0, x) dx − ρv · ∇ξ dx dτ = 0 Ω (B.1.9) 0 (B.1.10) Ω ∞ for all ξ ∈ D −∞, +∞; Cper . Identities (B.1.9) and (B.1.10) are exactly weak forms of the equations (B.1.1). Neither Oseen nor later on Leray [74] used the word ”weak” in their understanding of solution, but both of them work with it. While Oseen established the results † Oseen in his monograph [102] treats the Navier-Stokes fluids and their linearizations only. ‡ See also [34] and [35]. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 39 concerning local-in-time existence, uniqueness and regularity for large data, Leray [74] proved large-time and large-data existence for weak solutions of the NavierStokes equations, verifying thus (I), leaving as open the tasks (II) and (III). These tasks are still unresolved to our knowledge. The tasks (II) and (III) for the Navier-Stokes equation (B.1.8) represent the third millenium problem of the Clay Mathematical Institute [33]. The next issue concerns the function spaces where the solution satisfying (B.1.9) and (B.1.10) are to be found. There is an interesting link between the constitutive theory via the maximization of the entropy production presented in Chapter A and the choice of function spaces where weak solutions are constructed. We showed earlier how the form of the constitutive equation for the Cauchy stress can be determined knowing the constitutive equations for the specific Helmoltz free energy ψ and for the rate of dissipation ξ by maximizing w.r.t D’s fulfilling the reduced thermomechanical equation and the divergenceless condition as the constraint. Here, we show that the form of ψ determines function spaces for ρ, while the form of ξ determines the function space for v. This link would become even more transparent for more complex problems ([88] for example). Consider ψ and ξ of the form ψ = Ψ(ρ) and ξ = 2ν(p, ρ, D · D)D · D. (B.1.11) Assume that ρ fulfills 0 ≤ sup 0≤t≤T and 0≤ Z T 0 Z Z ρΨ(ρ(t, x)) dx < ∞, (B.1.12) Ω ν(p, ρ, D(v) · D(v))D(v) · D(v) dxdt < ∞. (B.1.13) Ω If for example Ψ(ρ) = ργ with γ > 1 and ν(p, ρ, D(v) · D(v)) = ν0 , then (B.1.12) and (B.1.13) imply that ρ ∈ L∞ (0, T ; Lγ+1 per ) and D(v) ∈ L2 (0, T ; L2per ). (B.1.14) In general, depending on specific structure of Ψ, (B.1.12) implies that ρ ∈ L∞ (0, T ; XΨ ) for some space XΨ . (B.1.15) $LaTeX: 2005/4/20 $ 40 J. Málek and K. R. Rajagopal If Ψ(ρ) = ργ , then XΨ = Lγ+1 per . Similarly, depending on the form of ν, one can conclude that D(v) ∈ Ydis or v ∈ Xdis for certain function spaces Ydis and Xdis , respectively. 1,2 (Ω)). In case of the constant viscosity Ydis = L2 (0, T ; L2per ) and Xdis = L2 (0, T ; Wper Note that the reduced thermomechanical equation (A.2.28) requires that T · D(v) = ξ + ρψ̇ = 2ν(p, ρ, D(v) · D(v))D(v) · D(v) + ρΨ̇(ρ). Now, if we formally set ϕ = v in (B.1.9) we obtain Z Z Z 1 d 2 %|v| dx + T · D(v) dx = %b · v dx 2 dt Ω Ω Ω (B.1.16) (B.1.17) and using Eq. (B.1.16) we see that the second term in (B.1.17) can be expressed as Z Z Z T(p, %, D(v)) · D(v)) dx = Ξ(p, %, D(v)) dx + %ψ̇ dx Ω Ω Ω Z Z d (%ψ) dx = Ξ(p, %, D(v)) dx + Ω dt Z ZΩ d 2 2 = %ψ dx , ν(p, %, |D(v)| ) |D(v)| dx + dt Ω Ω (B.1.18) where we used the fact that %̇ = 0 (see (B.1.1)). Assume that ρ0 and v0 are Ω-periodic functions satisfying (α1 , α2 being positive constants) % 0 ∈ XΨ v0 ∈ L2 (Ω) and α1 ≤ %0 ≤ α2 , and div v0 = 0. (B.1.19) (B.1.20) Then the fact that ρ fulfills the transport equation implies that α1 ≤ ρ(x, t) ≤ α2 for all (x, t) ∈ Ω × (0, +∞). (B.1.21) Consequently, it follows from (B.1.17)-(B.1.18) and (B.1.19)-(B.1.20) that (for all T > 0) v ∈ L∞ (0, T ; L2per ) ∩ Xdis and ρ ∈ L∞ (0, T ; XΨ ). (B.1.22) Specific description depends on the behavior of the viscosity with respect to D, p and ρ respectively. See Subsection 7.3 for further details. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 41 1.4. Models and their invariance with respect to scaling Solutions of the equations for power-law fluids (B.1.7) considered for r ∈ (1, 3) are invariant with respect to the scaling r−1 2 vλ (t, x) = λ 3−r v(λ 3−r t, λx), r−1 (B.1.23) 2 pλ (t, x) = λ2 3−r p(λ 3−r t, λx). It means that if (v, p) solves (B.1.7) with b = 0, then (v λ , pλ ) solves (B.1.7) as well. Note that NSEs are also included by setting r = 2 in (B.1.7). Applying this scaling we may magnify the flow near the point of interest located inside the fluid domain. Studying the behavior of the averaged rate of dissipation d(v) defined through d(v) := Z 0 −1 Z ξ(D(v)) dx dt = 2ν1 B1 (0) Z 0 −1 Z |D(v)|r dx dt (B.1.24) B1 (0) for d(vλ ) as λ → ∞, we can give the following classification of the problem: → 0 supercritical, if d(vλ ) → A ∈ (0, ∞) as λ → ∞ then the problem is critical, →∞ subcritical. Roughly speaking, we may say that for a subcritical problem the zooming (near possible singularity) is penalized by d(vλ ) as λ → ∞, while for supercritical case the energy dissipated out the system is insensitive measure of this magnification. Because of this, standard regularity techniques based on difference quotient methods should in principle works for subcritical case, while supercritical problems are difficult to handle without any additional information and they are even difficult to treat since weak formulation are not suitable for the application of finer regularity techniques. The Navier-Stokes equations in three spatial dimension represent a supercritical problem. In order to overcome a drawback of ”supercritical” problems to fully exploit fine regularity techniques, Caffareli, Kohn and Nirenberg [16] introduced the notion of suitable weak solution, and established its existence. A key new property of thus suitable form of weak solution is the local energy inequality. For (B.1.1) this is formally achieved by taking a sum of two identities: the first one is obtained by setting ϕ = vφ in (B.1.9) and the second one by setting ξ = |v|2 2 φ $LaTeX: 2005/4/20 $ 42 J. Málek and K. R. Rajagopal ∞ in (B.1.10), with φ ∈ D(−∞, +∞; Cper ) satisfying φ(x, t) ≥ 0 for all t, x. The local energy inequality thus reads Z Z tZ 1 (ρ|v|2 φ)(x, t) dx + (T · D(v)φ) (x, τ ) dx dτ ≤ 2 Ω 0 Ω Z Z Z 1 t 1 ρ|v|2 (v · ∇φ + φ,t ) (x, τ ) dx dτ (B.1.25) ρ0 (x)|v0 (x)|2 φ(0, x) dx + 2 Ω 2 0 Ω Z tZ − (Tv · ∇φ − ρb · vφ) (x, τ ) dx dτ. 0 Ω Using this tool, Caffarelli, Kohn and Nirenberg [16] were able to give a significant improvement in characterizing the structure of possible singularities for the NavierStokes equations in three dimensions. (Section 6 addresses this issue.) Since in three spatial dimension d(vλ ) defined through (B.1.24) fulfils d(vλ ) = 2ν1 λ 5r−11 3−r Z 0 − 12 λ 3−r Z |D(v1 )|r dy dτ, r ∈ (1, 3) (B.1.26) B1/λ (0) we see that the evolutionary equations for power-law fluids represent a subcritical 11 5 . Thus, the power-law fluids model should be mathematically ≥ 11 5 . The same is true for the Ladyzhenskaya’s equations (B.1.6) problem as r > treatable for r which has in comparison with the power-law fluid model one better property: the viscosity ν(D) = ν0 + ν1 |D|r−2 and consequently the corresponding nonlinear operator − div((ν0 + ν1 |D(v)|r−2 )D(v)) are not degenerate (while for power-law fluid, ν(D) = ν1 |D|r−2 as |D| → 0 degenerates for r > 2, and becomes singular for r < 2). The Ladyzhenskaya’s equations with r ≥ 11 5 are mathematically self-consistent; large-time existence of weak solution (task (I)) has been proved by Ladyzhenskaya (see [72], [65]), she also established large-time uniqueness (task (II)) for r ≥ Task (II) and (III) for r ≥ 11 5 5 2. were completed by Bellout, Bloom and Nečas in [8], Málek, Nečas and Růžička in [83] and by Málek, Nečas, Rokyta and Růžička in [81], although the boundedness of the velocity is perhaps explicitly stated in this contribution for the first time. The results in [83] give however the most difficult steps in this direction. Sections 3,4 and 5 focus on this topic. Mathematical self-consistency of the Ladyzhenskaya’s equations (and some of its generalization) is the central issue of this contribution. After introducing the notion of weak solution and suitable weak solution to equations for fluids with shear dependent viscosity (B.1.4) that includes the NSEs, Ladyzhenskaya’s equations and power-law fluids as special cases, we deal with large-time existence of these models $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 43 in Section 3 and using two methods we establish the existence of suitable weak solution for r > 59 , and the existence of weak solution satisfying only global energy inequality for r > 85 . Regularity of such solution is studied in Section 4 and established for r ≥ Particularly, if r ≥ 11 5 11 5 . we conclude that the velocity is bounded. We also outline how the higher differentiability technique can be used as a tool in the existence theory. Uniqueness and large-time behavior is addressed in Section 5. The short Section 6 gives a survey of the results dealing with structure of possible singularities of flows for the Navier-Stokes fluid. The final Section 7 states briefly results on large-time and large-data existence for further models namely for homogeneous fluid with pressure dependent viscosity and for inhomogeneous fluids with density or shear dependent viscosity. 2. Definitions of (suitable) weak solutions Before giving a precise formulation of (suitable) weak solution to the system of PDEs (B.1.4)-(B.1.5) describing unsteady flows of fluids with shear dependent viscosity, we need to specify the assumptions characterizing the structure of the tensor S = 2ν(|D(v)|2 )D(v), and to define function spaces we work with. 2.1. Assumptions concerning the stress tensor Let us compute the expression 3 X ∂Sij (A) ∂S(A) · (B ⊗ B) := Bij Bkl ∂A ∂Akl i,j,k=1 for the Cauchy stress of Ladyzhenskaya’s fluids and for power-law fluids. In the case of Ladyzhenskaya’s fluid, i.e., when we have S(A) = 2 ν0 + ν1 |A|r−2 A (r > 2) (B.2.1) ∂S(A) · (B ⊗ B) = 2 ν0 + ν1 |A|r−2 |B|2 + 2ν1 (r − 2)|A|r−4 (A · B)2 , ∂A (B.2.2) which implies ∂S(A) · (B ⊗ B) ≥ 2 ν0 + ν1 |A|r−2 |B|2 ≥ C1 1 + |A|r−2 |B|2 ∂A (B.2.3) with C1 = 2 min(ν0 , ν1 ) $LaTeX: 2005/4/20 $ 44 J. Málek and K. R. Rajagopal and ∂S(A) · (B ⊗ B) ≤ C2 1 + |A|r−2 |B|2 with C2 = 2 max(ν0 , ν1 (r − 1)) , (B.2.4) ∂A while for power-law fluids (set ν0 = 0 in (B.2.2)) 2ν1 |A|r−2 |B|2 ∂S(A) · (B ⊗ B) ≥ 2ν (r − 1)|A|r−2 |B|2 ∂A 1 if r ≥ 2 (B.2.5) if r < 2 and 2ν1 (r − 1)|A|r−2 |B|2 ∂S(A) · (B ⊗ B) ≤ 2ν |A|r−2 |B|2 ∂A 1 if r ≥ 2 (B.2.6) if r < 2. Motivated by the inequalities (B.2.3)-(B.2.4) for Ladyzhenskaya’s fluid and (B.2.5)(B.2.6) for the power-law fluid we put the following assumption on S. Let κ be either 0 or 1. We assume that 3×3 S : R3×3 → R3×3 with S(0) = 0 fulfils S ∈ C 1 R3×3 , (B.2.7) and there are two positive constants C1 and C2 such that for a certain r ∈ (1, ∞) 3×3 and for all 0 6= A, B ∈ Rsym ∂S(A) C1 κ + |A|r−2 |B|2 ≤ · (B ⊗ B) ≤ C2 κ + |A|r−2 |B|2 . ∂A (B.2.8) We also use convention that κ = 0 if r < 2. 2.2. Function spaces We primarily deal with functions defined on R3 that are periodic with the pe∞ riodic cell Ω = (0, L)3 . The space Cper consists of smooth Ω-periodic functions. Let r be such that 1 ≤ r < ∞. The Lebesgue space Lrper is introduced as the R ∞ closure of Cper -functions with Ω f (x) dx = 0 where the closure is made w.r.t. the R 1,r norm k · kr , where kf krr = Ω |f (x)|r dx. The Sobolev space Wper is the space of Ω-periodic Lebesgue-measurable functions f : R3 → R such that ∂xi f exists in 1,r a weak sense and f and ∂xi f belong to Lrper . Both Lrper and Wper are Banach 1 1 R R r r spaces with the norms kf kr := Ω |f (x)| dx and kf k1,r := Ω |∇x f (x)|r dx r , respectively. Let (X, k · kX ) be a Banach space of scalar functions defined on Ω. Then X 3 represents the space of vector-valued functions whose all components belong to X. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations Also, X ∗ denotes the dual space to X. For r 0 = 1,r ∗ instead of Wper . r r−1 , 45 −1,r we usually write Wper 1,r 1,r We also introduce the space Wper,div being a closed subspace of Wper 3 0 defined as the closure (w.r.t the norm k · k1,r ) of all smooth Ω-periodic functions v with the 1,r 1,r zero mean value such that div v = 0. Note that Wper,div = {v ∈ Wper , div v = 0}. If Y is any Banach space, T ∈ (0, ∞) and 1 ≤ q ≤ ∞, then Lq (0, T ; Y ) denotes the Bochner space formed by functions g : (0, T ) → Y such that, for 1 ≤ q < ∞, q1 R T is finite. The norm in L∞ (0, T ; Y ) is defined as kgkLq (0,T ;Y ) := 0 kg(t)kqY dt infimum of supt∈[0,T ]\E kg(t)kY , where infimum is taken over all subsets E of [0, T ] having zero Lebesgue measure. Also, if X is a reflexive Banach space, then Xweak denotes the space equipped with the weak topology. Thus, for example n o C(0, T ; Xweak ) ≡ ϕ ∈ L∞ (0, T ; X); hϕ(·), hi ∈ C(h0, T i) for all h ∈ X ∗ . Let 1 < α, β < ∞. Let X be a Banach space, and let X0 , X1 be separable and reflexive Banach spaces satisfying X0 ,→,→ X ,→ X1 . Then the Aubin-Lions lemma [76] says that the space W := v ∈ Lα (0, T ; X0); v,t ∈ Lβ (0, T ; X1) is compactly embedded into Lα (0, T ; X), i.e., W ,→,→ Lα (0, T ; X). 2.3. Definition of Problem (P) and its (suitable) weak solutions Our main task is to study the mathematical properties of Problem (P) consisting of: • four partial differential equations (B.1.4) with S satisfying (B.2.7)-(B.2.8), • the spatially periodic requirement (A.4.6), • the initial condition in R3 . v(0, ·) = v0 (·) (B.2.9) Let T > 0 be fixed, but arbitrary number. We assume that given functions b and v0 fulfil 1,r b ∈ Lr (0, T ; Wper ) and ∞ ∗ div v0 = 0 in (Cper ) ∗ 0 0 −1,r = Lr (0, T ; Wper ), and v0 ∈ L2per . (B.2.10) (B.2.11) $LaTeX: 2005/4/20 $ 46 J. Málek and K. R. Rajagopal Let (B.2.7) and (B.2.8) hold. We say that (v, p) = (v1 , v2 , v3 , p) is a suitable weak solution to Problem (P) provided that 5r 5r 1,r v ∈ C(0, T ; L2weak (Ω)) ∩ Lr (0, T ; Wper,div ) ∩ L 3 (0, T ; L 3 ); 0 0 0 11 −1,r 0 Lr (0, T ; Wper Lr (0, T ; Lrper ) , for r ≥ ) 5 v,t ∈ and p ∈ 5r −1, 5r 11 L 5r 6 6 (0, T ; W L 5r 6 (0, T ; L 6 ) ) per for r < ; per 5 lim kv(t) − v0 k22 = 0; t→0+ Z (B.2.12) (B.2.13) (B.2.14) T hv,t (t), ϕ(t)i − ((v ⊗ v)(t), ∇ϕ(t)) + (S(D(v(t))), D(ϕ(t))) 0 −(p(t), div ϕ(t)) dt = Z T 1,s for all ϕ ∈ Ls (0, T ; Wper ) hb(t), ϕ(t)i dt 0 5r 6 11 11 and s = if <r< ; 5 5r − 6 5 5 Z tZ Z 1 2 |v| φ (t, x) dx + S(D(v)) · D(v)φ dx dτ 2 Ω 0 Ω Z t Z Z tZ 1 |v|2 2 ≤ φ,t dx dτ + hb, vφi dτ |v0 (x)| φ(0, x) dx + 2 Ω 0 0 Ω 2 Z tZ 2 |v| + v + pv − S(D(v))v · ∇φ dx dτ 2 0 Ω (B.2.15) with s = r if r ≥ (B.2.16) ∞ valid for all φ ∈ D(−∞, +∞; Cper ), φ ≥ 0 and for almost all t ∈ (0, T i. 5r 5r • If r ≥ 3, the assertion v ∈ L 3 (0, T ; L 3 ) in (B.2.12) can be improved. For example, if r > 3, v being in Lr (0, T ; W 1,r ) implies v ∈ Lr (0, T ; C 0, r−3 3r ). Since our interest is focused on r ∈ (1, 3) we do not discuss this alternative in what follows. • Note that (B.2.12) and (B.2.13) ensure that all terms in (B.2.15) have sense for r > 65 , while all terms in (B.2.16) are finite if r > 59 . • Note that taking φ ≡ 1, one can conclude from (B.2.16) the standard energy inequality 1 kv(t)k22 + 2 Z t 0 1 (S(D(v)), D(v)) dτ ≤ kv0 k22 + 2 Z t hb, vi dτ (B.2.17) 0 has sense provided that the right hand side is finite. We say that (v, p) is weak solution to Problem (P) if (B.2.12)–(B.2.15) and (B.2.17) hold. • For the NSEs, local energy inequality takes slightly different form due to the linearity of S that allows to perform the integration per parts once more. This $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 47 gives 1 2 Z 2 |v| φ (t, x) dx + ν0 Ω 1 ≤ 2 Z Z tZ 0 Z 2 |∇v|2 φ dx dτ Ω tZ pv · ∇φ dx dτ + |v0 (x)| φ(0, x) dx + 0 Ω Ω Z tZ |v|2 + (φ,t + ν0 4φ + v · ∇φ) dτ 0 Ω 2 Z t hb, vφi dτ (B.2.18) 0 ∞ valid for all φ ∈ D(−∞, +∞; Cper ), φ ≥ 0 and for almost all t ∈ (0, T i. • Note that for r ≥ 11 5 we can set ϕ = v or ϕ = vφ in (B.2.15) and derive (B.2.16) 11 5 , we have v ∈ C(0, T ; L2per ) 1,r 1,r ∗ that follows from the fact that v ∈ Lr (0, T ; Wper ) and v,t ∈ Lr (0, T ; Wper ) . and (B.2.17) in the form of equality. Also, for r ≥ • Notice that the assumption (B.2.8) holds for the Ladyzhenskaya’s equations with κ = 1, and for power-law fluids with κ = 0. 2.4. Useful inequalities We first obtain several inequalities that are consequences of (B.2.7) and (B.2.8). Since (S(A) − S(B)) · (A − B) = = Z Z 1 d Sij (B + s(A − B))ds (A − B)ij ds 1 ∂Sij (B + s(A − B))(A − B)kl (A − B)ij ds, ∂Akl 0 0 it follows from the first inequality in (B.2.8) that Z 1 (κ + |B + s(A − B)|r−2 )ds|A − B|2 ≥ 0. (B.2.19) (S(A) − S(B)) · (A − B) ≥ C1 0 If r ≥ 2, (B.2.19) then implies (see Lemma 5.1.19 in [81] or [23]) |A − B|r if κ = 0, (S(A) − S(B)) · (A − B) ≥ C3 |A − B|2 + |A − B|r if κ = 1. Consequently, for A = D(u) and B = D(v) we have kD(u − v)krr ((S(D(u)) − S(D(v)), D(u − v)) ≥ C3 kD(u − v)k2 + kD(u − v)kr 2 r (B.2.20) if κ = 0, if κ = 1. (B.2.21) If r < 2 (and κ = 0), we show below that (B.2.19) implies (S(D(u)) − S(D(v)), D(u − v)) kD(u) + D(v)kr2−r ≥ C4 kD(u − v)k2r . (B.2.22) $LaTeX: 2005/4/20 $ 48 J. Málek and K. R. Rajagopal Consequently, setting v ≡ 0 in (B.2.21) and (B.2.22) we conclude (S(D(u)), D(u)) ≥ C5 kD(u)krr . (B.2.23) In fact it directly follows from (B.2.19) that S is strictly monotone, i.e. (S(A) − S(B)) · (A − B) > 0 if A 6= B. Also, if D(u), D(v) ∈ Lr (0, T ; Lr (Ω)3×3 ), (B.2.22) and Hölder’s inequality lead to Z T 0 kD(u − v)krr dt Z ≤ C6 T (S(D(u)) − S(D(v)), D(u − v)) dt 0 (B.2.24) ! r2 To see (B.2.22) we use Hölder inequality and inequality |B + s(A − B)| ≤ |A| + |B| valid for all s ∈ (0, 1) in the following calculation, where D(s) abbreviates D(v) + sD(u − v): kD(u − v)krr = = Z Z Ω ≤c Z h 1 Z |D(u) − D(v)|r dx Ω |D(s)|r−2 ds |D(u − v)|2 0 r2 Z 1 |D(s)|r−2 ds 0 (S(D(v)) − S(D(u))) · (D(u − v)) dx ≤ Z h i r2 Z Z Ω Ω (S(D(v)) − S(D(u))) · (D(u − v)) dx Ω −r 2 i r2 dx 1 |D(s)| r−2 0 2−r kD(u) + D(v)kr 2 ds −r 2−r dx 2−r 2 r which implies (B.2.22). Analogously, we could check that for r ∈ (1, 2) and θ ∈ ( 1r , 1) it holds for D(u), D(v) ∈ Lr (0, T ; Lr (Ω)3×3 ) Z T 0 kD(u − v)krθ rθ dt ≤ C̃6 Z T 0 Z [S(D(u)) − S(D(v)) · D(u − v))]θ dx dt Ω ! r2 (B.2.25) r where C̃6 depends on |Ω| and T , and the L -norms of D(u) and D(v). It also follows from (B.2.7) and (B.2.8): S(A) = S(A) − S(0) = ≤ C2 Z 1 Z 1 0 d S(sA) ds = ds Z 1 0 ∂S(sA) · A ds ∂A (κ + sr−2 |A|r−2 ) ds |A| ≤ C2 κ|A| + C2 0 1 |A|r−1 . r−1 $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 49 Consequently, using the convection that κ = 0 if r < 2, we have |S(A)| ≤ C2 κ|A| + C2 1 |A|r−1 ≤ C0 (κ + |A|)r−1 . r−1 (B.2.26) If r ∈ (1, +∞), Korn’s inequality states, see [98] or [91] that there is CN > 0 such that 1,r k∇ukr ≤ CN kD(u)kr for all u ∈ Wper or W01,r . (B.2.27) 3. Existence of a (suitable) weak solution 3.1. Formulation of the results and bibliographical notes The aim of this section is to establish the following result on large-time and large-data existence of suitable weak solution to unsteady flows of fluids with shear dependent viscosity. Theorem 3.1. Assume that S : R3×3 → R3×3 satisfies (B.2.7) and (B.2.8) with fixed parameter r. Let also b and v0 fulfil (B.2.10) and (B.2.11), respectively. If r> 8 , 5 (B.3.1) then there is a weak solution to Problem (P). If in addition r> 9 , 5 (B.3.2) then there is a suitable weak solution to Problem (P). Finally, if r≥ 11 , 5 (B.3.3) then the local energy equality hold and v ∈ C(0, T ; L2per ). Mathematical analysis of (B.1.4)-(B.1.5) was initiated by Ladyzhenskaya (see [72], [65]) who prove the existence of weak solution for r ≥ 11 5 treating homogeneous Dirichlet (i.e. no-slip) boundary condition. Her approach based on a combination of monotone operator theory together with the compactness arguments works for easier boundary-value problems, as spatial periodic problem or Navier’s slip as well. Provided that the viscosity depends on the full velocity gradient, i.e. ν = ν(|∇v|2 ), the same results are presented in the book of Lions [76]. For a complementary reading, see Kaniel [55]. $LaTeX: 2005/4/20 $ 50 J. Málek and K. R. Rajagopal The following table gives an overview of the results and methods available in two and general dimensions. We discuss below the results in three dimensions in detail. Spatially-periodic problem monotone operators + compactness monotone operators ∞ +L -function regularity technique d=2 d≥3 ⇒ r≥2 r ≥1+ ⇒ r> 3 2 r≥ 2(d + 1) d+2 [43] ⇒ r>1 r> 3d d+2 [81] ? [58] (higher-differentiability) C 1,α − regularity ⇒ Dirichlet problem (No-slip boundary) monotone operators + compactness regularity technique r≥ 4 3 ⇒ r≥2 ⇒ r≥ (higher-differentiability) C 1,α − regularity ⇒ 3 2 r≥2 r ≥1+ Refs. 2d d+2 2d d+2 r ≥ 2 (d = 3) ? [65],[76] [72], [65], [76] [57], [82] [56]. Note that Theorem 3.1 covers also the large-time and large-data existence for the NSEs, the results obtained for the Cauchy problem in the fundamental article by Leray [74], and extended to bounded domains with no-slip boundary conditions by Hopf [52] and to notion of suitable weak solution by Caffarelli, Kohn and Nierenberg [16]. The technique of monotone operators [65] or [76] however does not cover these results (as (B.3.3) does not include r = 2). This gap in the existence theory was filled by the result presented by Málek, Nečas, Rokyta and Růžička in [81], see [8] and [83] for the first appearance. The method based on the regularity technique to obtain fractional higher differentiability gives, among other results, the existence of weak solution for r fulfilling (B.3.2). This concerns the spatially periodic problem (A.4.6). For no-slip boundary conditions, the existence for r ≥ 2 is established in [82]. The idea of this method will be explained in Section 4. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 51 Later on in [43], using the facts that the nonlinear operator is strictly monotone and only L∞ -test function are permitted, Frehse, Málek and Steinhauer extended in some sense the existence theory for non-linear parabolic systems with L1 -right hand side performed for example in [11] and proved the existence of weak solution for r > 58 . In the following subsection the proof of Theorem 3.1 is established using the approach from [65] for r ≥ 11 5 and from [43] for r ∈ ( 85 , 11 5 ). Note that the last result for r ∈ ( 58 , 2) for no-slip boundary conditions is not completely solved yet. Frehse and Málek conjecture in [42] that one can exploit the restriction that only Lipschitz test function are admissible and establish the existence of weak solution for r > 65 . See [44] for details on this technique for time independent problems. 3.2. Definition of an approximate Problem (P ε,η ) and apriori estimates R Let η > 0 and ε > 0 be fixed. If u ∈ L1loc (R3 ), then u∗ω η := η13 R3 ω x−y u(y) dy η R with ω(·) ∈ D(B1 (0)), ω ≥ 0, ω being radially symmetric, B1 (0) ω = 1, is the standard regularization of a function u. $LaTeX: 2005/4/20 $ 52 J. Málek and K. R. Rajagopal We consider Problem (P ε,η ) to find (v, p) := (vε,η , pε,η ) such that† div v + ε|p|α p = 0 2−r α = r−1 with α = 5r − 12 6 11 , 5 11 for r < , 5 for r ≥ v,t + div ((v ∗ ω η ) ⊗ v) − div S(D(v)) = −∇p + b, (B.3.10) (B.3.11) and (vε,η , pε,η ) are Ω-periodic functions fulfilling (A.4.6) and each vε,η starts with the initial value specified in (B.2.9). Note that (B.3.10)-(B.3.11) is tantamount to 1 α+1 div v 1 v,t + div ((v ∗ ω ) ⊗ v) − div S(D(v)) − ∇ = b, (B.3.12) α ε | div v| α+1 η with p := − 1 ε 1 α+1 α | div v|− α+1 div v defined after solving for v = vε,η (B.3.12) together with (B.2.9) and (A.4.6). This kind of approximation is in the literature called quasi-compressible approximation or the problem with penalized divergenceless constraint. Although in (B.3.12) three non-linear operators appear, the solvability of (A.3.10) is not difficult due to the fact that the first operator div ((v ∗ ω η ) ⊗ v) is compact, the second and third one (for ε > 0 fixed) are monotone and the fol† There is a clear hint regarding the choice of α. Applying formally div to (B.1.4) 2 with b = 0, we obtain p = (−4)−1 div div (v ⊗ v − S(D(v))) . (B.3.4) Since the energy inequality implies that 1,r v ∈ L∞ (0, T ; L2per ) ∩ Lr (0, T ; Wper ), 2(3r−q) q(5r−6) the interpolation inequality kukq ≤ kuk2 v∈L 5r 3 3r q−2 kuk q3r5r−6 3−r (B.3.5) (for r < 3) leads to 5r 3 (0, T ; Lper ). (B.3.6) Consequently v⊗v ∈L Due to (B.2.26), S(D(v)) behaves as |∇v|r−1 5r 6 5r 6 ). (0, T ; Lper (B.3.7) and thus 0 0 S(D(v)) ∈ Lr (0, T ; Lrper ). (B.3.8) It thus follows from (B.3.4), (B.3.7) and (B.3.8) that p ∈ Lq (0, T ; Lqper ) If r ≥ 11 , 5 then q = r r−1 while for r < 11 , 5 where q = min q= 5r . 6 5r r , 6 r−1 . (B.3.9) In both cases, we choose α in such way that α + 2 = q. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 53 lowing estimates are available: εkpkα+2 α+2 1 d 2 1 kvk2 + S(D(v)) · D(v) dx + α+1 α+2 1 2 dt Ω k div vk α+1 α+2 ε α+1 Z (B.3.13) = hb, vi ≤ kbk−1,r0 k∇vkr . Note that α+2= r0 5r 6 , r 1 r − 1 and α+1= 5r − 6 6 α+2 = α+1 11 5 . for 11 r< 5 r≥ 5r 5r − 6 (B.3.14) Inequality (B.2.23) and Korn’s inequality (B.2.27) then allow us to conclude from (B.3.13) that sup kvε,η (t)k22 + t∈[0,T ] Z T 0 k∇vε,η (t)krr dt + ε Z T 0 kpε,η (t)kα+2 α+2 dt+ 1 1+α Z T α+2 1 + k div vε,η (t)k α+1 α+2 dt ≤ K, ε α+1 0 (B.3.15) where K is an absolute constant depending on kbkLr0 (0,T ;W −1,r0 ) , kv0 k2 and the per constant C0−1 from (B.2.26). It follows from the first two terms (implying that vε,η belongs to L∞ (0, T ; L2per )∩ 1,r Lr (0, T ; Wper ) uniformly w.r.t. ε and η) that for r ∈ (1, 3) Z T 5r kvε,η k 5r3 dt ≤ K. (B.3.16) 3 0 Due to (B.3.15) and (B.2.26) we also have Z T 0 kS(D(v))krr0 dt ≤ K. (B.3.17) 0 Looking at the equation (B.3.12) and using the estimates (B.3.15)–(B.3.17) we 0 0 −1,r see that the third and fifth term belong to Lr (0, T ; Wper (Ω)) (even uniformly 0 η −1,r 0 w.r.t. ε and η), the second term div (v ∗ ω ) ⊗ v) belongs to Lr (0, T ; Wper (Ω)) uniformly w.r.t. ε and η for r ≥ 11 5 , 5r 11 5 −1, 5r it belongs to L 6 (0, T ; Wper 6 (Ω)) −α 1 (again uniformly w.r.t. ε and η). The term (ε)− α+1 div | div v| α+1 (div v) I also 0 0 for r < −1,r belongs to Lr (0, T ; Wper (Ω)) for r ≥ 11 5 , 5r −1, 5r 6 and to L 6 (0, T ; Wper (Ω)) otherwise, however not uniformly w.r.t. ε > 0. $LaTeX: 2005/4/20 $ 54 J. Málek and K. R. Rajagopal As a consequence of this consideration, we have vε,η ,t ∈ 0 −1,r 0 Lr (0, T ; Wper ) −1, 5r 6 L 5r 6 (0, T ; W ) per 11 , 5 11 for r < . 5 for r ≥ (B.3.18) −α 1 Obviously, if we eliminate the term (ε)− α+1 div | div v| α+1 (div v) I using diver- genceless test functions we obtain the estimates that are uniform w.r.t. ε. Doing so, we obtain Z T 0 α+2 kvε,η ,t kW −1,α+2 dt ≤ K. (B.3.19) per,div 3.3. Solvability of an approximative problem In this subsection ε and η are fixed, and thus we write (v, p) instead (v ε,η , pε,η ). Based on (B.3.15) we set X := 1,r Lr (0, T ; Wper ) 11 , 5 11 . if r < 5 if r ≥ 5r 5r {u ∈ Lr (0, T ; W 1,r ); div u ∈ L 5r−6 (0, T ; L 5r−6 )} per Let {ωs }∞ s=1 be a basis of X. We construct a solution to (B.3.10)-(B.3.11), more precisely to (B.3.12) via Galerkin approximations {vN }∞ N =1 being of the form vN (t, x) = N X s cN s (t)ω (x), s=1 ∞ where cN := {cN s (t)}s=1 solve the system of ordinary differential equations: d N s (v , ω ) − ((vN ∗ ω η ) ⊗ vN , ω s ) + (S(D(vN )), D(ω s )) dt (B.3.20) −α 1 + 1 (| div vN | α+1 div vN , div ωs ) = hb, ω s i for s = 1, 2, . . . , N . ε α+1 Due to linearity of the second component in all expressions, we obtain (B.3.13) for vN that leads to (B.3.15), (B.3.16) and (B.3.17) for v N . Local-in-time existence of solution to (B.3.20) follows from Caratheodory theory, global-in-time existence is then consequence of (B.3.15), or its variant for v N . It also follows from (B.3.15), N ∞ (B.3.17) and (B.3.18) that there is a subsequence {vn }∞ n=1 ⊂ {v }N =1 and v ∈ $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 0 55 0 X ∩ L∞ (0, T ; L2per ), S ∈ Lr (0, T ; Lr (Ω)3×3 ) and P ∈ Lα+2 (0, T ; Lα+2) such that vn * v vn,t * v,t S(D(vn )) * S n n P (v ) := | div v | −α α+1 n div v * P weakly in X *-weakly in L∞ (0, T ; L2per ), −1,α+2 weakly in Lα+2 (0, T ; Wper ), 0 weakly in L (B.3.22) 0 (B.3.23) (0, T ; Lα+2 per ), (B.3.24) weakly in Lr (0, T ; Lr (Ω)3×3 ), α+2 (B.3.21) and thanks to Aubin-Lions compactness lemma (cf. [81], Lemma 1.2.48 or [76], Section 1.5) vn → v strongly in Lr (0, T ; Lqper ) for all q ∈ h1, 3r ). 3−r (B.3.25) Simple arguments then lead to the conclusion that v, S and P fulfil (for almost all t ∈ (0, T i) Z ( t 0= 0 ) 1 α+1 1 hv,t , ϕi − ((v ∗ ω ) ⊗ v, ∇ϕ) + (S, D(ϕ)) + (P , div ϕ) − hb, ϕi dτ ε (B.3.26) η for all ϕ ∈ X. Particularly, for ϕ = v we have # 1 α+1 Z t" 1 1 2 2 0 = (kv(t)k2 − kv0 k2 ) + (P , div v) − hb, vi dτ. (S, D(v)) + 2 ε 0 (B.3.27) Since for ψ ∈ X: 0≤ Z t (S(D(vn ))−S(D(ψ)), D(vn )−D(ψ))+ 0 1 α+1 1 (P (vn )−P (ψ), div(vn −ψ)) ε we use the equation (B.3.13) with vn instead of v to replace the term Z t (S(D(vn )), D(vn )) + 0 1 α+1 1 (P (vn ), div vn )dτ ε and pass to the limit as n → ∞. Using (B.3.21)-(B.3.25) we conclude # 1 α+1 Z t" 1 0≤ (S − S(D(ψ)), D(v) − D(ψ)) + (P − P (ψ), div(v − ψ)) dτ ε 0 (B.3.28) for all ψ ∈ X. A possible choice ψ = v ± λϕ, λ > 0, and continuity of the operators in (B.3.28) $LaTeX: 2005/4/20 $ 56 J. Málek and K. R. Rajagopal (for λ → 0+ ) then imply 0= Z t (S − S(D(v)), D(ϕ)) + 0 1 α+1 1 (P − P (v), div ϕ) dτ, ε (B.3.29) that says ) 1 α+1 1 (P , div ϕ) dτ (S, D(ϕ)) + ε ) 1 α+1 Z t( 1 = (S(D(v)), D(ϕ)) + (P (v), div ϕ) dτ. ε 0 Z t( 0 and (B.3.26) leads to the equation for (v, p) = (v ε,η , pε,η ) Z t hv,t , ϕi − ((v ∗ ω η ) ⊗ v, ∇ϕ) + (S(D(v)), D(ϕ)) dτ 0 + 1 Z t α+1 1 ε 0 | valid for all ϕ ∈ X. | div v| {z =:−p −α 1+α div v, div ϕ dτ = } Z t hb, ϕi dτ (B.3.30) 0 3.4. Further uniform estimates w.r.t ε and η Recall first that {vε,η } fulfil (B.3.15), (B.3.17) and (B.3.19) where K is independent of ε and η. Next, we focus on the uniform estimates of the pressure pε,η := − 1 −α 1 ε α+1 | div vε,η | 1+α div vε,η . We start with taking in (B.3.30) ϕ = ∇hε,η , where hε,η solves 4h ε,η = |p ε,η α ε,η 3 | p in R , h ε,η being Ω-periodic, Z hε,η (x) dx = 0 (B.3.31) Ω with the estimate khε,η k2,q ≤ Ckpε,η kα+1 (α+1)q q ∈ (1, ∞), (B.3.32) where C is independent of ε and η, but it may depend on q and L. As the result we obtain Z t Z t ε,η ε,η α+2 kp kα+2 dτ ≤ hvε,η idτ ,t , ∇h 0 0 Z tZ + |vε,η ||vε,η ∗ ω η ||∇(2) hε,η |dx dτ 0 Ω Z tZ |S(D(vε,η ))||D(∇hε,η )|dx dτ := I1 + I2 + I3 . + 0 (B.3.33) Ω $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 57 Terms I2 and I3 and (B.3.32) suggest to set q such that q(α + 1)-norm for p on right hand side of (B.3.32) equals to (α + 2)-norm of p on the left hand side of (B.3.33). This gives q := α+2 α+1 and it is easy to check that using (B.3.16) and (B.3.17), I2 and I3 are then controlled† Focusing on I1 , we observe first that vε,η can be decomposed into the sum v vε,η = vε,η div + ∇g ε,η (Helmholtz decomposition) (B.3.34) where ε,η div vε,η div = 0, v vε,η div and g and −4g ε,η Thus, gv ε vε,η = − div v ε,η = ε|p ε,η α ε,η | p are Ω − periodic Z , ε,η gv (B.3.35) dx = 0. (B.3.36) Ω = −hε,η as follows from (B.3.31) and (B.3.36) and unique solvability of the Laplace equation in the considered class of functions. Consequently Z t Z t Z t vε,η g,t ε,η vε,η ε,η , ∇(−hε,η )i dτ hvε,η h∇g , ∇h idτ = −ε h∇ , ∇h i dτ = ,t ,t ε 0 0 0 Z t vε,η g,t ε,η ε,η 1 1 d = −ε k∇ k22 dτ = − (kg v (t)k22 − kg v (0)k22 ) ≤ 0 2 dt ε 2ε 0 (B.3.37) ε,η as 4g v (0) = div vε,η (0) = 0 ⇒ g v ε,η (0) = 0. (The reader may wish to perform ε,η this argument how to treat the term I1 first for smooth approximations (vε,η m , pm ) and then pass to the limit as m → ∞, whereas vε,η m follows from the density of 1 α 1,r α+1 | div vε,η |− α+1 div vε,η .) smooth functions in Lr (0, T ; Wper ) and pε,η m := −(ε) m m Thus, it follows from (B.3.33)-(B.3.37) that† Z T kpε,η kα+2 α+2 dτ ≤ K. (B.3.38) 0 Consequently, we can strengthen (B.3.19) to conclude from (B.3.30) that Z T α+2 (B.3.39) kvε,η ,t kW −1,α+2 dτ ≤ K. per 0 † To be more explicit, considering for example the term I3 we have Z t 1 Z t Z α+1 α+2 α+2 α+2 |I3 | ≤ kS(D(vε,η ))kα+2 ≤ KC k∇2 hε,η k α+1 α+2 dτ α+2 dτ 0 0 α+1 t 0 kpε,η kα+2 α+2 dτ α+1 α+2 , where we used the fact that α + 2 ≥ r 0 for arbitrary r > 1. † Note that this step can be repeated without any change for the Navier’s boundary conditions, it is however open in general for no-slip boundary conditions due to the fact that ∇h is not an admissible function. $LaTeX: 2005/4/20 $ 58 J. Málek and K. R. Rajagopal The estimates (B.3.38) and (B.3.39) are uniform w.r.t. η. If we however relax this requirement and use the fact that v ∗ ω η is a smooth function for η > 0 fixed we obtain, proceeding as above, Z T 0 and Z T 0 kpε,η krr0 dτ ≤ C(η −1 ) r kvε,η ,t k 0 −1,r Wper 0 0 (B.3.40) dτ ≤ C(η −1 ). (B.3.41) 3.5. Limit ε → 0 For fixed η > 0, we establish in this section the existence of (suitable) weak solution to the problem div v = 0, v,t + div((v ∗ ω η ) ⊗ v) − div(S(D(v))) = −∇p + b Z Z p dx = 0 for i = 1, 2, 3 vi dx = vi , p Ω − periodic with (P η ) Ω Ω v(0, ·) = v0 in Ω, if the parameter r appearing in (B.2.8) fulfils r> 8 . 5 (B.3.42) Using the estimates (B.3.15), (B.3.17), (B.3.40) and (B.3.41) uniform w.r.t. ε > 0, an letting ε → 0 we can find a sequence {vn , pn } chosen from {vε,η , pε,η }, and a limit element {v, p} := {vη , pη } such that vn * v vn,t * v,t vn → v 1,r weakly in Lr (0, T ; Wper ) and *-weakly in L∞ (0, T ; L2per ) 0 0 −1,r weakly in Lr (0, T ; Wper ) Lr (0, T ; Lq ) for all q ∈ h1, 3r ) per 3−r strongly in Ls (0, T ; Lsper ) for all s ∈ h1, 5r ) 3 S(D(vn )) * S 0 0 weakly in Lr (0, T ; Lrper ) (B.3.43) (B.3.44) (B.3.45) (B.3.46) and pn * p 0 0 weakly in Lr (0, T ; Lrper ). (B.3.47) It also follows from the fourth term in (B.3.15) (as ε → 0) that div v = 0 a.e. in (0, T ) × Ω. (B.3.48) $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 59 Consider (B.3.30) for {vn , pn } instead of {v, p} = {vε,η , pε,η }, we can pass to the limit as n → ∞ and obtain with help of (B.3.43)-(B.3.47) Z tn 0 o hv,t , ϕi − ((v ∗ ω η ) ⊗ v, ∇ϕ) + (S(D(v)), D(ϕ)) − (p, div ϕ) dτ Z t 1,r hb, ϕidτ for all ϕ ∈ Lr (0, T, Wper ). = (B.3.49) 0 provided that we show that S = S(D(v)) a.e. in (0, T ) × Ω. (B.3.50) For this purpose, we consider again (B.3.30) for (v n , pn ) and set ϕ = vn −v therein. Then Z T 0 hvn,t − v,t , vn − vidt + =− Z T Z T (S(D(vn )) − S(D(v)), D(vn − v))dt 0 hv,t , vn − vidt + (S(D(v)), D(vn − v))dt + hb, vn − vidt 0 − Z T ((vn ∗ ω η ) ⊗ vn , ∇(vn − v))dt. (B.3.51) 0 Let n → ∞. The first term on the right hand side of (B.3.51) vanishes due to RT weak convergence (B.3.43), the last integral that equals to 0 (div(vn ∗ ω η )vn , vn − RT v)dt + 0 ((vn ∗ ω η ) ⊗ (vn − v), ∇vn ) also vanishes since (B.3.45) and |∇vn | |vn | is uniformly integrable if r > 58 . Consequently, using (B.3.51) and (B.2.20) resp. (B.2.21), we have† lim n→∞ Z T 0 kD(vn ) − D(v)krr dt = 0, (B.3.52) Thus D(vn ) → D(v) a.e. in (0, T )×Ω (at least for subsequence) and Vitali’s Lemma (see [81] Lemma 2.1 or Dunford and Schvartz [27]) and (B.3.17) give S(D(vn )) → S(D(v)) a.e. in (0, T ) × Ω that implies (B.3.50). † We also used the fact that Z T n hvn ,t − v,t , v − vidt = 0 1 n 1 1 kv (T ) − v(T )k22 − kvn (0) − v(0)k22 = kvn (T ) − v(T )k22 . 2 2 2 This requires to check that vn (0) = v(0) = v0 . We skip it however here and show it later for more difficult case. $LaTeX: 2005/4/20 $ 60 J. Málek and K. R. Rajagopal ∞ Taking ϕ = vφ, φ ∈ D(−∞, ∞; Cper ) in (B.3.49) we conclude the local energy equality Z tZ Z 1 |v|2 φ (t, x) dx + S(D(v)) · D(v)φ dx dτ 2 Ω 0 Ω Z Z Z Z t 1 1 t = |v0 (x)|2 φ(0, x) dx + |v|2 φ,t dx dτ + hb, vφi dτ 2 Ω 2 0 Ω 0 Z tZ 2 |v| η + (v ∗ ω ) + pv − S(D(v))v · ∇φ dx dτ 2 0 Ω (B.3.53) Also taking ϕ = v in (B.3.49) we have global energy equality Z t Z t 1 1 kv(t)k22 + (S(D(v)), D(v)) dτ = kv0 k22 + hb, vidτ 2 2 0 0 (B.3.54) and thanks to lower-semicontinuity of the norms w.r.t. weak convergence it follows from (B.3.15),(B.3.16), (B.3.17), (B.3.38) and (B.3.39) that (v, p) = (vη , pη ) fulfils the following estimates that are uniform w.r.t. η > 0: Z T Z T 5r kvη k 5r3 dt ≤ K, k∇vη krr dt + sup kvη (t)k22 + t∈[0,T ] Z Z Z T 0 kS(D(vη ))krr0 dt ≤ K, 0 T 0 T 0 kpη kα+2 α+2 dt ≤ K (B.3.56) with α + 2 = kvη,t kα+2 dt ≤ K. W −1,α+2 r0 5r 6 11 5 , 11 5 11 if r < 5 if r ≥ (B.3.57) (B.3.58) per 3.6. Limit η → 0, the case r ≥ If r ≥ (B.3.55) 3 0 0 11 5 the available uniform estimates coincides with those needed to pass to the limit in Subsection 3.5. Thus, we proceed as above. The quadratic convective term requires vn → v strongly in L2 (0, T ; L2 (Ω)). (B.3.59) This follows from the Aubin-Lions lemma provided that r> 6 , 5 which is of course trivially fulfilled here (and also in the next Subsection). The other argument coincides with those used in Subsection 3.5. For r ≥ 11 5 , we have thus the existence of weak solution fulfilling energy equality, local energy equality, etc. The proof of Theorem 3.1 in the case r ≥ 11 5 is complete. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 3.7. Limit η → 0, the case 8 5 <r< 61 11 5 5r 5r 3 We start observing that if u ∈ L 3 (0, T ; Lper ) and ∇u ∈ Lr (0, T ; Lrper ) then [∇u](u ∗ ω η ) ∈ L1 (0, T ; L1per ) (B.3.60) uniformly w.r.t. η > 0 provided that r≥ Thus, introducing for r > 8 5 8 . 5 (B.3.61) and δ ∈ (0, 85 (r − 85 )) the space of divergenceless functions 1,r Xδ := {ϕ ∈ Lr (0, T ; Wper,div )∩L 1+δ δ 1+δ δ )} , (0, T ; Lper (B.3.62) and using the fact that −((v ∗ ω η ) ⊗ v, ∇ϕ) = ([∇v](v ∗ ω η ), ϕ), (B.3.63) it follows from (B.3.49) that kvη,t kXδ∗ ≤ K uniformly w.r.t. η > 0. (B.3.64) Letting η tend to zero, and using (B.3.55)-(B.3.58), (B.3.64) and the Aubin-Lions compactness lemma, we find a subsequence {(vk , pk )}k∈N and ”its weak limit” {(v, p)} such that (r < 11 5 ) 1,r weakly in Lr (0, T ; Wper ) and *-weakly in L∞ (0, T ; L2per ), vk * v (B.3.65) −1, 5r 6 5r 6 )and in Xδ∗ , vk,t * v,t weakly in L (0, T ; Wper vk → v strongly in Lr (0, T ; Lqper ) for all q ∈ h1, (B.3.66) 3r ), 3−r 6 strongly in Ls (0, T ; L2per ) for all s > 1, if r > , 5 vk → v vk → v a.e. in (0, T ) × Ω, pk * p 6 ), weakly in L 6 (0, T ; Lper 5r 0 (B.3.67) (B.3.68) (B.3.69) 5r (B.3.70) 0 and there is S ∈ Lr (0, T ; Lrper ) so that S(D(vk )) * S 0 0 weakly in Lr (0, T ; Lrper ).. (B.3.71) $LaTeX: 2005/4/20 $ 62 J. Málek and K. R. Rajagopal Also, it follows from (B.3.68) that vk (t) → v(t) strongly in L2per for all t ∈ [0, T ] \ N, (B.3.72) where N has zero one-dimensional Lebesgue measure. In order to identify S with S(D(v)) we showed in previous sections that this follows from Z TZ (S(D(vn )) − S(D(v))) · (D(vn ) − D(v)) dx dt = 0 , lim n→∞ 0 (B.3.73) Ω using the fact that this integral operator is uniformly monotone (note that it would suffice to know that this operator is strictly monotone). Here, we will show a condition weaker than (B.3.73), namely: 1 for every ε∗ > 0 and for some θ ∈ ( , 1) there is a subsequence r (*) k ∞ {vn }∞ n=1 of {v }k=1 such that Z TZ h iθ (S(D(vn )) − S(D(v))) · (D(vn ) − D(v)) dx dt ≤ ε∗ . lim n→∞ 0 Ω Once (*) is proved, we take ε∗m → 0 and for each m ∈ N select gradually (not relabelled) subsequences so that the Cantor diagonal sequence (again not relabelled) fulfils D(vn ) → D(v) a.e. in (0, T ) × R3 . (B.3.74) Vitali’s theorem and (B.3.74) then imply S = S(D(v)) a.e. in (0, T ) × R3 . The convergences (B.3.65)-(B.3.71) and (B.3.74) clearly suffice to pass to the limit from the weak formulation of Problem (P η ) to the weak form of Problem (P). It remains to verify (*). For this purpose we set g k := |∇vk |r + |∇v|r + (|S(D(vk ))| + |S(D(v))|)(|D(vk )| + |D(v)|) . (B.3.75) Clearly g k ≥ 0 and 0≤ Z T 0 Z g k dx dt ≤ K. (K > 1) (B.3.76) Ω We prove the following property (K is referred to (B.3.76)) for every ε∗∗ > 0 there is L ≤ (**) ε∗∗ k ∞ , {vn }∞ n=1 ⊂ {v }k=1 and K sets E n := {(x, t) ∈ (0, T ) × Ω; L2 ≤ |vn (t, x) − v(t, x)| < L} such that Z g n dx dt ≤ ε∗∗ . E $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations To see it, we fix ε∗∗ ∈ (0, 1), set L1 = ε∗ K 63 and take N ∈ N such that for N ε∗∗ > K (K refers to (B.3.76)). Defining iteratively Li = L2i−1 for i = 2, 3, . . . N , we set E k,i = {(t, x) ∈ (0, T ) × Ω; L2i ≤ |vk (t, x) − v(t, x)| < Li } (i = 1, 2, . . . N.) For k ∈ N fixed, E k,i are mutually disjoint. Consequently, N Z X i=1 g k dx dt ≤ K. E k,i As N ε∗∗ > K, for each k ∈ N there is i0 (k) ∈ {1, . . . N } such that Z g k dx dt ≤ ε∗∗ . E k,i0 (k) However, i0 (k) are taken from finite set of indices. Then, there has to be a sequence {vn } ⊂ {vk } such that i0 (n) = i∗0 for each n (i∗0 ∈ {1, 2, . . . N } fixed). The property ∗ (**) is then proved setting L = Li∗0 and E n = E n,i0 . Returning to our aim to verify (*), we consider (vn , pn ) satisfying (B.3.49), (B.3.53), (B.3.54) and having all convergence properties stated in (B.3.65)-(B.3.71) and (**), and we set ϕ in (B.3.49) of the form n |v − v| ,1 − ∇z n , ϕn := hn − ∇z := (vn − v) 1 − min L (B.3.77) where L comes from (**), and z n solves for n |v − v| ,1 = div hn f n := div (vn − v) 1 − min L the problem −4z n = f n z n being Ω-periodic , Z z n dx = 0. (B.3.78) Ω We summarize the properties of hn , z n and ϕn . Introducing Qn through Qn := {(t, x) ∈ (0, T ) × Ω; |vn (t, x) − v(t, x)| < L} we note first that hn = 0 on (0, T ) × Ω \ Qn , (B.3.79) for all (t, x) ∈ (0, T ) × Ω. (B.3.80) and |h(t, x)| ≤ L Consequently, owing to (B.3.69) and Lebesgue’s theorem we have for all s ∈ h1, ∞) Z T 0 khn kss dt → 0 as n → ∞, (B.3.81) $LaTeX: 2005/4/20 $ 64 J. Málek and K. R. Rajagopal and due to Ls -theory for the Laplace-operator, it follows from (B.3.78) that Z T k∇z n kss → 0 0 as n → ∞. (B.3.82) From (B.3.81) and (B.3.82) it follows (ϕn defined in (B.3.77)) Z T 0 kϕn kss dt → 0 as n → ∞. (B.3.83) Next, (χZ denotes the characteristic function of a set Z) f n = div hn = (vn − v) · (vn − v)j ∇(vn − v)j χQ n . L |vn − v| Splitting Qn into E n (introduced in (**)) and its complement, and using the fact that |f n |r ≤ |∇(vn − v)|r ≤ |∇vn |r + |∇v|r on E n and |f n |r ≤ L(|∇vn |r + |∇v|r ≤ ε∗ n r K (|∇v | + |∇v|r ) on Qn \ E n we conclude from (**) that Z T 0 kf n krr,Qn dt ≤ 2ε∗ , (B.3.84) and using Lr -regularity for the Laplace operator and (B.3.78) Z T k∇(2) z n krr,Qn dt ≤ 2Creg ε∗ . (B.3.85) 1,r weakly in Lr (0, T ; Wper ) and also in Xδ , (B.3.86) 0 Note also that ϕn * 0 where Xδ is defined in (B.3.62). Inserting ϕn of the form (B.3.77) into (B.3.49) we obtain (note that the term with pressure vanishes as div ϕn = 0) Z T 0 hvn,t n − v,t , ϕ i dt + Z T =− − (S(D(vn )) − S(D(v)), D(ϕn )) dt 0 Z Z T 0 T 1 ((vn ∗ ω n )[∇vn ], ϕn ) dt + 0 hv,t , ϕn i dt − Z Z T hb, ϕn i dt (B.3.87) 0 (S(D(v)), D(ϕn )) dt. 0 It is not difficult to see that all terms on the right hand side of (B.3.87) tend to 0: the first one due to (B.3.60) and (B.3.83), the third one due to the fact that v,t ∈ Xδ∗ and (B.3.86) holds, and the second and fourth terms also due to (B.3.86). $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations Let H : h0, ∞) → R satisfies H(0) = 0 and H 0 (s) = (1 − min( 65 √ s L , 1)). Then the first term on the left hand side is non-negative as Z T Z T hvn,t − v,t , hn i dt = H(|vn − v|2 (T )) ≥ 0. hvn,t − v,t , ϕn i dt = 0 0 We thus conclude from (B.3.87) that Z (S(D(vn )) − S(D(v)))(D(vn ) − D(v))) dx dt lim n→∞ Qn Z (B.3.88) (|S(D(vn ))| + |S(D(v))|) |∇(vn − v)| + |∇(2) z n | dx dt . ≤ lim n→∞ Qn Arguing analogously asin the derivation of Z (S(D(vn )) − S(D(v)))(D(vn ) − D(v))) dx dt ≤ Cε∗ lim n→∞ Since Z 0 T (B.3.89) Qn Z h Ω iθ (S(D(vn )) − S(D(v)))(D(vn − v)) dx dt Z Z [. . .]θ dx dt [. . .]θ dx dt + = (0,T )×Ω\Qn Qn Hölder ≤ Z + [. . .] dx dt Qn Z θ |Qn |1−θ [. . .] dx dt (0,T )×Ω\Qn !θ |{(t, x); |vn − v| > L}|1−θ ≤ C ∗ ε∗ , where we apply (B.3.89) to handle the first term and the convergence in measure to treat the second term, we finally conclude that (*) holds. 3.8. Continuity w.r.t. time in weak topology of L2per With all convergences established in previous sections, particularly with (B.3.65)(B.3.70) and (B.3.74), it is straightforward to conclude that (v, p) fulfils weak identity (B.2.15). Taking here ϕ of the form 1,s ϕ(τ, x) = χht0 ,ti (τ )ϕ̃(x), where ϕ̃ ∈ Wper and t0 , t ∈ h0, T i, we obtain (v(t), ϕ̃) − (v(t0 ), ϕ̃) = Z t (v(τ ) ⊗ v(τ ), ∇ϕ̃) − (S(D(v)), D(ϕ̃)) t0 + hb(τ ), ϕ̃i + (p(τ ), div ϕ̃) dτ. $LaTeX: 2005/4/20 $ 66 J. Málek and K. R. Rajagopal This implies (for r > 65 ) |(v(t), ϕ̃) − (v(t0 ), ϕ̃)| ≤ c Z t t0 kv(τ )k25r + k∇v(τ )krr−1 + kb(τ )k−1,r0 3 + kp(τ )kα+2 dτ kϕ̃k1,s . (B.3.90) Using also Hölder’s inequalities over time, we have 5r6 Z t 5r 5r−6 3 5r |(v(t), ϕ̃) − (v(t0 ), ϕ̃)| ≤ c |t − t0 | kv(τ )k 5r dτ 3 t0 + |t − t0 | + |t − t0 | 1 r 1 r Z Z α+1 + |t − t0 | α+2 t t0 t t0 Z k∇v(τ )krr dτ 0 kb(τ )kr−1,r0 t t0 r10 dτ r10 kp(τ )kα+2 α+2 dτ 1 α+2 kϕ̃k1,s . (B.3.91) Since all integrals are finite, (B.3.91) leads to the conclusion that (v(·), ϕ̃) is con1,s (Ω). In other words, tinuous at t0 for all ϕ̃ ∈ Wper 1,s ∗ v ∈ C(0, T ; (Wper )weak ) (B.3.92) or lim (v(t) − v(t0 ), ϕ̃) = 0 t→t0 1,s for all ϕ̃ ∈ Wper and for all t0 ∈ h0, T i (B.3.93) 1,s Since v ∈ L∞ (0, T ; L2per ) and Wper is dense in L2per , we see that v ∈ C(h0, T i; L2weak ), which is (B.2.12)1 . 3.9. (Local) Energy equality and inequality If r ≥ 11 5 , (B.2.15) permits to take ϕ = v or ϕ = vφ which implies both energy equality and its local version. If 11 5 > r > 58 , we take lim supn→∞ of (B.3.54) where v means vn , and t ∈ h0, T i \ N with N introduced in (B.3.72). Since • lim sup{an + bn } ≥ lim sup{an } + lim inf {bn }, n→∞ n→∞ n→∞ Z tZ Z tZ • S(D(v)) · D(v) dx dτ ≤ lim inf S(D(vn )) · D(vn ) dx dτ, 0 • vn0 n→∞ Ω 0 Ω n = v (0) for all n ∈ N, • (B.3.72) and (B.3.65) hold, $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 67 we see that energy inequality (B.2.17) directly follows. Similarly we argue letting n → ∞ in (B.3.53). Here we in addition need to pass to the limit in terms Z tZ 0 Ω |vn |2 n 1 (v ∗ ω n ) + pn vn 2 · ∇φ dx dτ, that follows from (B.3.68), (B.3.69) and (B.3.55)3 provided that r> 9 . 5 3.10. Attainment of the initial condition The property (B.2.14) is an easy consequence of energy inequality (B.2.17) and the following operations kv(t) − v0 k22 = kv(t)k22 + kv0 k22 − 2(v(t), v0 ) = kv(t)k22 − kv0 k22 − 2(v(t) − v0 , v0 ) Z t (B.2.17) ≤ −2 [(S(D(v)), D(v)) − hb, vi] dτ − 2(v(t) − v0 , v0 ). 0 (B.3.94) Letting t → 0+ in (B.3.94) we conclude (B.2.14) from (B.2.12)1 and the fact that (S(D(v)), D(v)) − hb, vi ∈ L1 (0, T ). 4. On smoothness of flows 4.1. A survey of regularity results The alternate topic of this section can be higher differentiability of weak solution (v, p) of Problem (P). For simplicity, we set b = 0. Since we deal with spatially periodic problem we are free of technical difficulties due to localization. For j = 1, 2, 3, ej denotes the basis vector in R3 (ej = (δ1j , δ2j , δ31 ), δij being the Kronecker delta). Let δ0 > 0 be fixed. Introducing for h ∈ (0, δ0 ) the notation j 4hj z(t, x) = z [+he ] (t, x) − z(t, x) := z(t, x + hej ) − z(t, x), $LaTeX: 2005/4/20 $ 68 J. Málek and K. R. Rajagopal it is not difficult to observe that (B.2.15) implies Z h i j S(D(v))[+he ] − S(D(v)) · D(ϕ) dx (B.4.1) Ω j = (4hj v) ⊗ v[+he ] , ∇ϕ + v ⊗ 4hj v, ∇ϕ + 4hj p, div ϕ , h[4hj v],t , ϕi + 1,s ) with s = r if r ≥ that holds for all ϕ ∈ Ls (0, T ; Wper 11 5 and s = 5r 5r−6 if 6 5 ≤r< 11 5 almost everywhere in (0, T ). It is a direct consequence of these requirements on ϕ 11 5 . in (B.4.1) and (B.2.15) that we can put ϕ = 4hj v in (B.4.1) only if r ≥ In order to relax such an apriori bound on r, we can use instead of (B.4.1) the weak formulation of Problem (P η ). Then for (v, p) = (vη , pη ) we have Z h i j S(D(v))[+he ] − S(D(v)) · D(ϕ) dx = 4hj p, div ϕ (B.4.2) Ω j + ((4hj v) ∗ ω η ) ⊗ v[+he ] , ∇ϕ + (v ∗ ω η ) ⊗ 4hj v, ∇ϕ h[4hj v],t , ϕi + 1,r valid for a.a. t ∈ (0, T ) and for all ϕ ∈ Lr (0, T ; Wper ). In (B.4.2), we are allowed to take ϕ = 4hj v in (B.4.2), having the same restriction on r needed for the solvability of (B.4.2), i.e., r > 85 . We aim to obtain higher differentiability estimates uniformly w.r.t. η > 0. Inserting ϕ = 4hj v into (B.4.2), noting that div 4kj v = 0 implying that the term involving the pressure as well as the last term appearing in (B.4.2) vanish, we obtain j j 1 d k4hj vk22 + S(D(v))[+he ] − S(D(v)), [D(v)][+he ] − D(v) 2 dt = − (4hj v ∗ ω η ) ⊗ 4hj v, ∇v[+he j ] (B.4.3) ≤ k∇vkr k4hj vk22r . r−1 Using (B.2.21) for r ≥ 2 (or (B.2.22) for r < 2), it follows from (B.4.3) that 1 d k4hj vk22 + k4hj D(v)k22 + k4hj D(v)krr ≤ k∇vkr k4hj vk22r . r−1 2 dt (B.4.4) If one concludes from (B.4.4) some higher differentiability estimates (even fractional ones suffice), the compact embedding theorem (the Aubin-Lions lemma) then lead to almost everywhere convergence for the velocity gradients. We can then pass to limit, as η → 0, from (B.3.49) to (B.2.15). The higher differentiability estimates thus represent another method to establish the large-time and large-data existence of weak solution to Problem (P). Carrying on the original contribution by Málek, Nečas and Růžička [83] and Bellout, Bloom and Nečas [8], see also [81], where however smoother approximations $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 69 of Problem (P) are considered†, it seems very likely that the outlined procedure is workable and one can thus find (fractional) higher differentability estimates for r > 95 . It is of interest to mention that this range for r’s coincides with that required for the existence of suitable weak solution. To be more precise, the following results are in place (see [83], [8], [81], [84]). 11 5 Theorem 4.1. (i) If r ≥ then there is a weak solution (v, p) to Problem (P) fulfilling sup k∇v(t)k22 t∈h0,T i Z Z T (κk∇ 0 T 2 vk22 + kD(v)kr N 2 ,r r (Ω) + k∇vkr3r ) kv,t k22 dt + sup k∇v(t)krr dt t∈h0,T i 0 sup kv,t k22 t∈h0,T i κ Z T 0 k∇v,t k22 dt + Z T 0 Z Ω |D(v)|r−2 |D(v,t )|2 dx dt ≤ C(k∇v0 k2 ) (B.4.5) ≤ C(k∇v0 kr ) (B.4.6) ≤ C(kv0 k2,q ), (B.4.7) where κ = 0 or 1 according to (B.2.8), q > 3 and kzkN α,r := sup 0<h≤δ0 Z Ω |z(x + h) − z(x)|r dx hαr r1 . (ii) if r ∈ h2, 11 5 ) then there is a weak solution (v, p) to Problem (P) such that v fulfills κ Z T k∇ 0 2 2 3r−5 vk2 r+1 dt + Z T k∇vk 0 r2 (3r−5) 3(r2 −3r+4) W s,r dt < ∞ 2 s ∈ (0, ), r (B.4.8) (iii) if r ∈ ( 95 , 2) then there is (v, p) to Problem (P) such that v fulfills Z T r(5r−9) 2 +8r−9) k∇2 vkr(−r dt ≤ ∞. (B.4.9) 0 In particular, for the spatially-periodic problem described by the Navier-Stokes equations in three dimensions, it follows from Theorem 4.1 that (set r = 2 in † In [83], the estimates are derived directly for Galerkin approximations using smooth basis of functions. In [8] a multipolar fluid model is used as a smooth approximation. Both approximations thus allow us to differentiate the equations of the approximative problems. $LaTeX: 2005/4/20 $ 70 J. Málek and K. R. Rajagopal (B.4.8)) there is a weak solution (v, p) such that Z T 2/3 k∇2 vk2 dt < ∞, (B.4.10) 0 the result established by Foias, Guillopé and Temam [37]. Regarding a no-slip boundary condition, Málek, Nečas and Růžička [82] considered the case r ≥ 2 with κ = 1 and showed that: • if r ≥ 9 4 (and r < 3) then there is a weak solution that fulfils 6 2 2,2 ), v ∈ L 2−r (0, T ; W 2, r+1 (Ω))3 ∪ L2 (0, T ; Wloc Z T 0 v,t ∈ L2 (0, T ; L2(Ω)3 ), Z (1 + |D(v)|)r−2 |D(∇v)|2 dx dt ≤ K (B.4.11) for all Ω0 ⊂⊂ Ω, Ω0 • if r ∈ h2, 94 ) then Z T 2 2r−3 k∇2 vk 3 6r−1 dt ≤ K < ∞. 0 (B.4.12) r+1 Note that (B.4.12) implies (B.4.10) even for Dirichlet (no-slip) boundary value problem. Instead of proving (B.4.5)-(B.4.10) rigorously, we rather provide a cascade of formal inequalities that form however essence of correct arguments. Details and many extensions can be found† in [81], [82] and [24]. This cascade consists of three levels of inequalities, considering the energy inequality as level zero. Level 1: Differentiate (B.1.4)2 w.r.t. xs and scalarly multiply the result with ∂v ∂xs . Level 2: Multiply (B.1.4)2 with v,t . Level 3: Differentiate (B.1.4)2 w.r.t. time t and use v,t as the multiplier. For r ≥ 11 5 the procedure leads to (B.4.5)-(B.4.7). For the Navier-Stokes equa- tions inequality (B.4.5) is not available and there is a plenty of results in literature asking the question what are the conditions implying (B.4.5). The well-known are so-called Prodi-Serrin conditions‡ saying that (B.4.5) holds provided that v ∈ Lq (0, T ; Ls ) with 2 3 + ≤ 1, q s s ≥ 3. (B.4.13) If s > 3, the result is established by Serrin in [130]. The most interesting limiting case L∞ (0, T ; L3) has been covered recently by Escauriaza, Seregin and Šverák † Dealing with approximations different than those used in Section 3. ‡ Prodi asks for conditions implying uniqueness of weak solution, see [105]. It revealed that the criterion coincides with that for regularity. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 71 ([30], [31], [128]). Other regularity criteria are expressed in terms of the velocity gradient (see [7] for example), the vorticity (see [18]), the pressure ([9], [93], [129]), or just one component of the velocity ([95], [96]) or the velocity gradient (see [18], [30]). The result in [61] and [134] extends (B.4.13) to the class L2 (0, T ; BM O). The regularity criteria expressed in terms of eigenvalues and eigenfunctions of the symmetric part of the velocity gradient were established in [96], [97]. While fractional higher differentiability result, as that mentioned in (B.4.10), gives compactness of velocity gradient, say in all Lq (0, T ; Lqper ), q < 2, they do give any improvement on the regularity of the velocity or its gradient alone. In terms of our ”level” inequalities (Doering and Gibbon [25] talk about the ladder where each split bar corresponds to a level above) for the Navier-Stokes equations in three dimensions, it is not known how to make the first step from ground (level zero) to the first rail (level 1). It is however proved that once the level 1 is achieved (in fact (B.4.13) or other criteria suffice), L∞ (0, T ; L2per ) integrability of any spatial or time derivatives of any order is available, provided that data (v 0 and b) are smooth enough. For Ladyzhenskaya’s equations, or for Problem (P) with κ = 1, Theorem 4.1 states that if r ≥ 11 5 , the first three levels (B.4.5)-(B.4.9) (of the ladder) are accessi- ble. It is however open how to proceed to high levels. More precisely, using (B.4.7) and using (B.1.4) we rewrite Problem (P) as div(v ⊗ v) − div S(D(v)) + ∇p = −v,t ∈ L∞ (0, T ; L2per ), (B.4.14) and we observe that we can apply the higher differentiability technique to almost all time instants. Doing so we conclude that Theorem 4.2. If r ≥ 11 5 then there is (v, p) solution to Problem (P) (with κ = 1) such that sup k∇2 v(t)k22 + sup k∇v(t)kr t t N 2 ,r r (Ω) + sup k∇v(t)kr3r ≤ K < ∞. (B.4.15) t In particular, v is bounded in (0, T ) × R3 . Thus, the task (III) large-time and large-data regularity is in sense given in Subsection B.1.2 fulfilled. The question if then also ∇v is bounded or Hölder continuous has been however unanswered yet. $LaTeX: 2005/4/20 $ 72 J. Málek and K. R. Rajagopal In next subsections we formally establish (B.4.5)-(B.4.7), and also (B.4.15). We also discuss related results on local-in-time existence of solutions with integrable second derivatives. 4.2. A cascade of inequalities for Ladyzhenskaya’s equations In this part, we consider the Ladyzhenskaya’s equations (B.1.6). It means we deal with the system (B.1.4) where S(D(v)) = ν0 + ν1 |D(v)|r−2 D(v). Note also that (B.2.8) holds with κ = 1. In the sequel we sometimes use the specific structure of S, sometimes we refer to (B.2.8). • Derivation of (B.4.5). We formally differentiate (B.1.4) with respect to spatial variable xs and take scalar product of the result with ∂v ∂xs . After summing over s = 1, 2, 3 and integrating by parts we obtain Z Z 1 d ∂S(D(v)) ∂vk ∂vi ∂vi k∇vk22 + · D(∇v) ⊗ D(∇v) dx = − . (B.4.16) 2 dt ∂D ∂x s ∂xk ∂xs Ω Ω Using (B.2.8) we conclude that 1 d k∇vk22 + C1 k∇2 vk22 + C1 Jr (v) ≤ k∇vk33 , 2 dt where Jr (v) = Since Z (B.4.17) |D(v)|r−2 |D(∇v)|2 dx. Ω Jr (v) ≥ c∗ k∇vkr3r , (B.4.18) see [81], p. 227, and Jr (v) ≥ c∗∗ kD(v)kr N 2 ,r r , (B.4.19) see [84] for the proof, then (B.4.17) and the energy inequality (B.2.17) implies (B.4.5) if r ≥ 3. If r < 3, then we incorporate the interpolation inequalities, 3−r r−1 kzk3 ≤ kzkr 2 kzk3r2 2 r−1 (B.4.20) r 3r−2 kzk3 ≤ kzk2 3r−2 kzk3r , 3(1−α) and use the splitting k∇vk33 = k∇vk3α 3 k∇vk3 for α ∈ h0, 1i. Then, we obtain C1 1 d k∇vk22 + C1 k∇2 k22 + Jr (v) + C1∗ k∇vkr3r 2 dt 2 ≤ r−1 3α (3−r)+ 3r(1−α) 6(1−α) 3r−2 3α r−1 3r−2 k∇vkr 2 k∇vk2 k∇vk3r2 . (B.4.21) $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations Setting Q1 := 3α (3−r) 2 r 3(1−α) 3r−2 , δ = Q11 . + Young’s inequality with Q2 := 3α r−1 2 r 73 r−1 and Q3 := 3(1 − α) 3r−2 , we apply Requiring also that Q2 δ 0 = 1, i.e. Q2 + Q1 = 1, we obtain 1 d k∇vk22 + 2C1 k∇2 vk22 + C1 Jr (v) + C1∗ k∇vkrr ≤ ck∇vkrr k∇vk2λ 2 , 2 dt where λ := 2 3−r . 3r − 5 (B.4.22) (B.4.23) Since λ≤1⇔r≥ 11 , 5 we obtain (B.4.5) applying Gronwall lemma. • Derivation of (B.4.6). The scalar multiplication of (B.1.4) with v,t and the integration over Ω leads to kv,t k22 − (div S(D(v)), v,t ) + (v,t ⊗ v, ∇v) = 0 . (B.4.24) Using the specific form of S and the integration by parts we obtain kv,t k22 + ν0 d ν1 d k∇vk22 + kD(v)krr = (v,t ⊗ v, ∇v) 2 dt r dt 1 ≤ kv,t k22 + k |v| |∇v| k22 . 2 (B.4.25) Since the estimate of the last term can be established with help of W 1,3r ,→ L∞ Z |v|2 |∇v|2 dx ≤ kvk2∞ k∇vk22 ≤ C(sup k∇v(t)k22 )k∇vk23r , (B.4.26) Ω t we see that (B.4.6) follows after integrating (B.4.25) over time and applying (B.4.5). It is also possible to conclude from (B.4.24) that kv,t (0)k22 ≤ C(kv0 k2,q ) . (B.4.27) • Derivation of (B.4.7). A formal differentiation of (B.1.4) with respect to time, and a multiplication of the result with v,t leads to Z ∂S(D(v)) 1 d 2 kv,t k2 + · D(v,t ) ⊗ D(v,t ) dx = (v,t ⊗ v, ∇v,t ), 2 dt ∂D Ω since (p,t , div v,t ) = 0 and (v⊗v,t , ∇v,t ) = (v, ∇ |v,t |2 2 ) = −(div v, |v,t |2 2 ) (B.4.28) = 0. Using (B.2.8), and applying Hölder’s inequality to the right-hand side of (B.4.28) gives Z 1 d 2 2 |D(v)|r−2 |D(v,t )|2 dx kv,t k2 + κk∇v,t k2 + 2 dt Ω κ 2 (B.4.29) ≤ k∇v,t k2 + kvk2∞ kv,t k22 2 κ ≤ k∇v,t k22 + Ck∇vk23r kv,t k22 . 2 $LaTeX: 2005/4/20 $ 74 J. Málek and K. R. Rajagopal The Gronwall lemma and (B.4.27) completes the formal proof of (B.4.7). 4.3. Boundedness of the velocity Derivation of (B.4.15). Since r ≥ 11 5 , (B.4.5)-(B.4.7) hold. We proceed similarly as in obtaining (B.4.16), the term with the time derivative v ,t is however treated differently: (∇v,t , ∇v) = −(v,t , 4v) ≤ kv,t k2 k∇2 vk2 . Using then the Hölder inequality and (B.4.18) we have instead of (B.4.17) C1 k∇2 vk22 + C1 Jr (v) + k∇vkr3r ≤ kv,t k22 + k∇vk33 . r−1 (B.4.30) 3−r The interpolation inequality kzk3 ≤ kzkr 2 kzk3r2 then gives C1 k∇2 v(t)k22 +C1 Jr (v(t))+C1 k∇vkr3r ≤ kv,t (t)k22 +(k∇v(t)krr ) 3 r−1 2r (k∇v(t)kr3r ) 3−r 2r . (B.4.31) As supkv,t (t)k22 < t and 3−r 2r < 1, we ∞ due to (B.4.7) and supk∇v(t)krr t 2,2 obtain (B.4.15). Since W < K ≤ ∞ owing to (B.4.6), 1 (Ω) ,→ C 0, 6 (Ω), we conclude from (B.4.15) that 1 v ∈ L∞ (0, T ; C 0, 6 (Ω)). (B.4.32) In particular, v is bounded in (0, T ) × Ω. 4.4. Fractional higher differentiability 5 3 Let <r< 11 5 (⇔ λ > 1). Since Jr (v) ≥ C11 2 2 r κk∇ vk2 + kD(v)k C k∇2 vk2r (1 + k∇vkr )2−r N 2 ,r r (Ω) if r ≥ 2 if r < 2, it follows from (B.4.22), see [81] for details, that κ Z 0 T kD(v)kr 2 ,r k∇2 vk22 N r (Ω) + dt ≤ ∞ (1 + k∇vk22 )λ (1 + k∇vk22 )λ if r ≥ 2 (B.4.33) if r < 2. (B.4.34) and Z T 0 k∇2 vk2r 1 dt ≤ ∞ (1 + k∇vkr )2−r (1 + k∇vk22 )λ $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 75 Hölder’s inequality and the energy inequality then leads to (B.4.8) and (B.4.10). Using such estimates, we can then apply interpolation inequalities to obtain fractional higher differentiability with the exponent greater than one. For example, for the Navier-Stokes equations we know from (B.4.8) and (B.2.12) that Z T Z T 2 3 dt < ∞. kvk21,2 dt < ∞ and kvk2,2 0 0 This then implies Z T 0 1 2 ≥ 1 ⇔ s ≤ , s ∈ h0, 1i. 2s + 1 2 2 2s+1 dt < ∞ kvk1+s,2 and 4.5. Short-time or small-data existence of ”smooth” solution Inequalities of the type (B.4.22) that can be rewritten in a simplified form y 0 (t) ≤ g(t)y(t)λ , where y(t) ≥ 0 and g ∈ L1 (0, T ), (B.4.35) serve, if λ > 1, as the key in proving either short-time and large-data or large-time and small-data existence of ”smooth” solution. Note that (B.4.22) takes the form of (B.4.35) with y(t) = k∇vk22 , g(t) = k∇vkrr 3−r and λ = 2 3r−5 , and the energy inequality (B.2.17) implies that for all T > 0 Z T 0 k∇vkrr ≤ ckv0 k22 (B.4.36) If λ > 1, (B.4.35) is tantamount to y(t) ≤ y(0) 1 (1 − I(t)(λ − 1)[y(0)]λ−1 ) λ−1 with I(t) := Z t g(τ ) dτ. (B.4.37) 0 and we observe that sup y(t) ≤ K < ∞ t provided that 1 − I(t)(λ − 1)[y(0)]λ−1 ≥ 1 . 2 (B.4.38) In the case of (B.4.22), the condition (B.4.38) reads Z t 2(λ−1) k∇vkrr dτ )k∇v0 k2 ≤ 1. 2(λ − 1)( (B.4.39) 0 It follows from (B.4.36) that (B.4.39) holds for all t > 0 provided that 2(λ−1) 2(λ − 1)ckv0 k22 k∇v0 k2 ≤ 1. (B.4.40) $LaTeX: 2005/4/20 $ 76 J. Málek and K. R. Rajagopal 1,2 Thus, if v0 ∈ Wper fulfils (B.4.40), there is a solution v such that for all T > 0 sup k∇v(t)k22 ≤ 2k∇v0 k22 . t∈(0,T ) Since Rt 0 k∇vkrr dτ → 0 as t → 0+ , it also follows from (B.4.37) and (B.4.39) that 1,2 for any v0 ∈ Wper there is t∗ > 0 such that weak solution v fulfils sup k∇v(t)k22 ≤ 2k∇v0 k22 . (B.4.41) t∈(0,t∗ ) In order to have an explicite bound on the length t∗ , one can proceed slightly differently starting again from the inequality (B.4.17). If we apply only the second interpolation inequality from the (B.4.20) we obtain 3r 1 d C1 6 r−1 3r−2 k∇vk22 + C1 k∇2 vk22 + Jr (v) + k∇vkr3r ≤ ck∇vk2 3r−2 k∇vk3r . (B.4.42) 2 dt 2 Since 3 3r−2 < 1 if and only if r > 53 , Young’s inequality leads to d 6 r−1 k∇vk22 + 2C1 k∇2 vk22 + C1 Jr (v) + k∇vkr3r ≤ ck∇vk2 3r−5 . dt (B.4.43) This is inequality of the type y 0 ≤ cy µ with µ = 3(r − 1) > 1. 3r − 5 Proceeding as above we observe that (B.4.41) holds provided that 0 < t∗ ≤ 1 2(µ−1) 2(µ − 1)k∇v0 k2 . To summarize, the following results follow from (B.4.22) (and a discussion above on level inequalities) for the Navier-Stokes equations and the Ladyzhenskaya’s equations. • three-dimensional flows driven by Navier-Stokes equations starting with smooth initial flow v0 are smooth on certain time interval (0, t∗ ). Also, if smooth initial condition v0 fulfils (B.4.40), large-time and small-data existence of smooth solution takes place. See [59] [76], [102], [22], [66], [77], [140] or [138]. • three-dimensional flows of power-law fluid or driven by Ladyzhenskaya’s equations ∗ with r ∈ ( 35 , 11 5 ) fulfil† (B.4.5)-(B.4.7) on certain (0, t ) for any smooth initial data. In particular, v is bounded on (0, t∗ ), Also, if v0 fulfils (B.4.40), large-time (and † Strictly speaking the inequalities (B.4.5)-(B.4.7) hold only for r ≥ 2. If r < 2, different norms appear in (B.4.5)-(B.4.7) (see [81], [87]). $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 77 small-data) existence of flows v fulfilling (B.4.5)-(B.4.7) and (B.4.15) is valid. Again v remains bounded provided v0 ∈ W 2,q , q > 3. Large-time existence of C(0, T ; W 2,q )-solutions for small data v0 ∈ W 2,q , q > 3 is also established in [2]. An improvement in the short-time and lasrge-data existence from the range r > 5 3 up to r > 7 5 is presented in [24]. 5. Uniqueness and large-data behavior The aim, to show internal mathematical consistency for Ladyzhenskaya’s equations if r ≥ 11 5 , will be completed by establishing two results on continuous dependence of flows on data, implying uniqueness. As a consequence, the asymptotic structure of all possible flows as t → ∞ can be studied. We present results on existence of exponential attractor. This is a compact set in the function space of initial conditions, invariant with respect to solution semigroup, having finite dimensional fractal dimension and attracting all trajectories exponentially. 5.1. Uniquely determined flows described by Ladyzhenskaya’s equations Theorem 5.1. Let (v1 , p1 ) and (v2 , p2 ) be two weak solutions to Problem (P) corresponding to data (v10 , b1 ) and (v20 , b2 ), respectively. If r≥ 5 2 (B.5.1) and 0 0 −1,r vi0 ∈ L2per and bi ∈ (Lr (0, T ; Wper )) (i = 1, 2), (B.5.2) then sup kv1 (t) − v2 (t)k22 ≤ h(v10 − v20 , b1 − b2 ) (B.5.3) t where h(ω0 , g, v20 , b2 ) := c1 kω0 k22 + Z T 0 0 kgkr(W 1,r )∗ per ! exp c2 kv20 k22 + Z T 0 0 kb2 kr(W 1,r )∗ per ! . Also, Z and T 0 k∇(v1 − v2 )k22 + k∇(v1 − v2 )krr dt ≤ ch(v10 − v20 , b1 − b2 ), Z T 0 0 k∇(p1 − p2 )krr0 dt ≤ ch(v10 − v20 , b1 − b2 ). (B.5.4) (B.5.5) In particular, Problem P is uniquely solvable in the class of weak solution. $LaTeX: 2005/4/20 $ 78 J. Málek and K. R. Rajagopal Proof: Taking the difference of (B.2.15) considered for (v 1 , p1 ) from (B.2.15) for (v2 , p2 ) we come, for r ≥ 11 5 , to the identity for ω = v2 − v1 and q = p2 − p1 hω ,t , ϕi + (S(D(v2 )) − S(D(v1 )), D(ϕ)) = (q, div ϕ) + hb2 − b2 , ϕi + (ω ⊗ v2 , ∇ϕ) + (v1 ⊗ ω, ∇ϕ) (B.5.6) 1,r valid for all ϕ ∈ Wper and a.a. t ∈ h0, T i. Taking ϕ = ω and observing (q, div ω) = 0 and (v1 ⊗ ω, ∇ω) = 0, (B.5.6) implies 1 d kωk22 + (S(D(v2 )) − S(D(v1 )), D(v2 − v1 )) 2 dt 1 2 (B.5.7) 2 = hb − b , ωi − (ω ⊗ ω, ∇v ). Monotone properties of S, i.e. (B.2.21) and (B.2.22), Korn’s inequality and duality estimates allow us to treat the term with b1 − b2 , then yield Z 0 1 d ν1 |ω|2 |∇v2 | dx. (B.5.8) kωk22 + ν0 k∇ωk22 + k∇ωkrr ≤ ckb2 − b1 kr(W 1,r )∗ + per 2 dt 2 Ω Using also Z 2r−3 r Ω |ω|2 |∇v2 | dx ≤ k∇v2 kr kωk22r ≤ k∇v2 kr kωk2 r−1 2r−3 3 kωk6r 3 ≤ ck∇v2 kr kωk2 r k∇ωk2r 2r ν0 ≤ k∇ωk22 + ck∇v2 kr2r−3 kωk22 , 2 it follows from (B.5.8) that h i d kωk22 + ν0 k∇ωk22 + ν1 k∇ωkrr dt 2r 0 ≤ c kb1 − b2 kr(W 1,r )∗ + k∇v2 kr2r−3 kωk22 . (B.5.9) per Neglecting the terms standing in square brackets, the Gronwall lemma completes then the proof of (B.5.3) provided that 2r 2r−3 ≤ r, which is exactly the condition 2r RT (B.5.1). The energy inequality (B.2.17) and (B.2.23) to estimate 0 k∇v2 kr2r−3 dt is also used. Integrating (B.5.9) over time between 0 and T , using (B.5.3) to control supt kωk22 lead then to (B.5.4). To conclude (B.5.5), we set ϕ := ∇h in (B.5.6), where h solves Z 2−r 2−r 1 |q| r−1 q 4h = |q| r−1 q − |Ω| Ω Z h dx = 0. h is Ω − periodic, (B.5.10) Ω $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 79 Then, 1 1 . kϕk1,r ≤ khk2,r ≤ ck |q| r−1 kr ≤ ckqkrr−1 0 (B.5.11) Since hω ,t , ϕi = hω ,t , ∇hi = 0, (B.5.6) with ϕ = ∇h then leads to Z 0 kqkrr0 ≤ c |∇ω||∇ϕ| + (|D(v1 )| + |D(v2 )|)r−2 |D(ω)||D(ϕ)| dx Ω 1 3r 3r k∇ϕkr + kb − b2 kW −1,r0 k∇ϕkr + kv1 + v2 k 2(2r−3) kωk 3−r (B.5.12) per 1 ≤ k∇ωk2 k∇ϕk2 + k∇ωkr k∇v + ∇v2 krr−2 k∇ϕkr + kb1 − b2 kW −1,r0 k∇ϕkr + kv1 + v2 k per 3r 2(2r−3) k∇ωkr k∇ϕkr . 5r 5r 1,r 3 )∩Lr (0, T ; Wper ), Using (B.5.11), Young’s inequality, the fact that v 1 +v2 ∈ L 3 (0, T ; Lper and finally (B.5.4), we obtain (B.5.5). Theorem 5.2. Let (v1 , p1 ) and (v2 , p2 ) be two weak solution to Problem (P) corresponding to data (v10 , b1 ) and (v20 , b2 ) respectively. If r≥ 11 , 5 (B.5.13) (v10 , b1 ) fulfills (B.5.2) and 1,2 v20 ∈ Wper (Ω) and b2 ∈ L2 (0, 2; L2per ), then for all t ∈ (0, T i the inequalities (B.5.3)-(B.5.5) hold with ! Z h(ω 0 , g, v20 , b2 ) := c1 kω0 k22 T + 0 0 kgkr(W 1,r )∗ per exp c2 kv20 k21,2 (B.5.14) + Z T 0 kb2 k22 ! . Consequently, a weak solution fulfilling in addition (B.4.5) is unique in the class of weak solutions. In another words, if data fulfils (B.5.14) Problem (P) is uniquely solvable. Proof: Since (v20 , b2 ) fulfils (B.5.14), Theorem 4.1 implies Z T Z T 2 r 2 2 k∇v k3r dt ≤ c(k∇v0 k2 + kb2 k22 dt). 0 0 Proceeding step by steps as in the proof of Theorem 5.1, we estimate the right hand side of (B.5.8) as follows: Z 2 2 2r−1 |ω|2 |∇v2 | dx ≤ k∇v2 k3r kωk2 6r ≤ k∇v2 k3r kωk2 2r k∇ωk2r Ω 3r−1 2r ν0 2r−1 ≤ k∇ωk22 + ck∇v2 k3r kωk22 . 2 $LaTeX: 2005/4/20 $ 80 J. Málek and K. R. Rajagopal As 2r 2r−1 ≤ r for r ≥ 11 5 , the remaining part of the proof coincides with that of Theorem 5.1. Uniqueness of weak solution of Problem (P) for r ≥ [76], uniqueness for r ≥ 11 5 5 2 is stated in [65], see also is mentioned in [78] and [81]. 5.2. Large-time behavior - the method of trajectories Not only are the Navier-Stokes equations the first system of nonlinear partial differential equations for which the methods of functional analysis were applied and developed†, the Navier-Stokes equations, at least in two dimension, serve also as the first system of equations of mathematical physics to which the theory of dynamical systems was addressed and further extended‡. The restriction to two dimensional flows is due to missing uniqueness and lack of regularity in three spatial dimensions. Owing to uniquely determined flows (v, p) of the Navier-Stokes fluid in two spatial dimensions, the mapping St : L2per → L2per such that St v0 = v(t) posses the semigroup property, i.e., S0 = Id and St+s = St Ss for all t, s ≥ 0. (B.5.15) We recall definitions of several basic notions. For later use, let (X, k · kX ) be a normed spaces and St : X → X having the properties (B.5.15). A bounded set B ⊂ X is said to be uniformly absorbing if for all B0 ⊂ X bounded there is t0 = t(B0 ) such that St B0 ⊂ B for all t ≥ t0 . A set B̃ ⊂ X is positively invariant w.r.t St if St B̃ ⊂ B̃ for all t ≥ 0. If there is a bounded set B ∗ ⊂ Y ,→,→ X that is uniformly absorbing all bounded sets in X and that is positively invariant, then A := \ [ St B ∗ s>0 t≥s is called global attractor as it shares the following properties: (i) A is compact in X, (ii) St A = A for all t ≥ 0, i.e., A is invariant w.r.t. St and (iii) A attracts all † We can refer to [74], [52], [76], [65], [137], [22], [73], etc. ‡ As general reference, we can give [63], [39], [68], [21], [139], [50], [29], [46], [69]. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 81 bounded sets of X, which means that¶ for all B ⊂ X bounded distX (St B, A) → 0 as t → ∞. Compactness of the global attractor recalls for the question of the finite dimension of large-time dynamics. For a compact set C ⊂ X, the fractal dimension dX f (C) is defined as dX f (C) := lim sup ε→0+ where NεX (C) log NεX (C) , log 1ε is the minimal number of ε-balls needed to cover C. According to Foias and Olson [38] if dX f (C) < m 2 ,m m Hölder continuous mapping from R ∈ N, then C can be placed into the graph of a onto C. This mapping is a projector if X is a Hilbert space. Thus the finiteness of the fractal dimension dX f (A) and its estimates from above (and even more importantly from below) give a worth characterization of large-time dynamics. The following elementary criterium holds (see [85], Lemma 1.3): Let (Y, k · kY ) ,→,→ (X, k · kX ) and C ⊂ X be bounded. (***) If there is L : X → Y, being Lipschitz continuous on C, and LC ⊂ C, then dX f (C) < ∞. To ensure an exponential rate of attraction, Eden, Foias, Nikolaenko and Temam [29] enlarge the global attractor and introduce the notion called exponential attractor. This is a subset of B ∗ having the following properties: (i) E is compact in X, (ii) E is positively invariant w.r.t. St , (iii) dX f (E) < ∞ and (iv) there are α1 , α2 > 0 such that distX (St B ∗ , E) ≤ α1 e−α2 t for all t ≥ 0. For two dimensional flows of the Navier-Stokes fluids, the existence of global (minimal B) attractor A ⊂ L2per was established by Ladyzhenskaya in [63]. Estimates on its fractal dimension were first studied by Foias and Temam [40], see also [68] for similar criterion. The up-to-date best estimates, based on the method of Lyapunov exponents, are due to Constantin and Foias (see [22] for example). A proof of the existence of exponential attractor is presented in [29]. It is natural to ask if the large-time dynamics of three-dimensional flows driven by the Ladyzhenskaya’s equations share the same properties as two-dimensional ¶ The Hausdorff distance distX (A, B) of two sets A, B ⊂ X is defined as distX (A, B) = sup inf kx − ykX . x∈A y∈B $LaTeX: 2005/4/20 $ 82 J. Málek and K. R. Rajagopal NSEs. The first result in this direction is due to Ladyzhenskaya [67] and [70] who proved the existence of global attractor for r ≥ 52 , leaving however open the quaestion of its dimension. Need to say that neither Ladyzhenskaya criterion requiring that orthonormal projectors commute with the nonlinear operator div(S(D(v))) nor the method of Lyapunov exponents requiring the (not available) regularity results for the linearized problem, can hardly be applicable to show finiteness of the fractal dimension of attractors. The criterium (***) cannot be also used for X = L2per 1,2 and Y = Wper . It is however elementary to verify (***) for X = L2 (0, `; L2per ) and 1,2 3,2 ∗ Y := {u ∈ L2 (0, `; Wper ), u,t ∈ L1 (0, `; (Wper ) )} where, ` > 0 is fixed. This is the first motivation to work with the set of `-trajectories rather than with single values v(t) ∈ L2per . The second motivation comes from uniqueness result 5 formulated in Theorem 5.2 for r ∈ h 11 5 , 2 ). We are not sure if just one trajectory starts from any v0 ∈ L2per (Theorem 5.2 says it is true if v0 is smoother, namely 1,2 v0 ∈ Wper ). However, once we fix any `-trajectory starting at v0 ∈ L2per , we know that it has uniquely defined continuation, as almost all values of the `-trajectory 1,2 belong to Wper . Thus the operators Lt : L2 (0, `; L2per ) → L2 (0, T ; L2per ), (B.5.16) that appends to any `-trajectory χ its uniquely defined shift at time t, have the semigroup property (B.5.15). Following Málek and Pražák [85], using the semigroup (B.5.16) it is not only possible to find A` ⊂ L2 (0, `; L2per ), the global attractor with respect to the semigroup Lt and with help of (***) to show that its fractal dimension is finite, but introducing A ⊂ L2per as set of all end-points of `-trajectories belonging to A` , it easily follows from Lipschitz (or at least Hölder) continuity of the mapping e : χ ∈ A` → χ(`) ∈ A that A is attractor with respect to original dynamics, with finite fractal dimension. The same approach gives also the existence of exponential attractor. Theorem 5.3. Let b ∈ L2per be time independent. Consider Problem (P) with r ≥ 11 5 and κ = 1 in (B.2.8), and with v0 ∈ L2per . Then this dynamical system possesses • a global attractor A ⊂ L2per with finite fractal dimension, • an exponential attractor E. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 83 X 2 In both cases, explicit upper bounds on dX f (A) and df (E) with X = Lper are avail- able. We refer to Málek and Pražák [85] for explanation of the method of trajectories, that originates in [78], and for the proof of Theorem 5.3. Explicit upper bounds on L2 df per (A) are given in [86]. See also [13] for a comparison of the estimates for twodimensional flows obtained by the method of Lyapunov exponents on one hand and the method (***) on the other hand. As the extreme case of the method of trajectories one can consider Sell’s study of ∞-trajectories of three-dimensional Navier-Stokes equations, see [126], suitable to treat ill-posed problem. It is worth of mentioning a Ladyzhenskaya’s counterexample to uniqueness of weak solutions to three-dimensional NSEs in a domain varying with time, see [64]. 6. On structure of possible singularities for flows of Navier-Stokes fluid It is hardly possible to cover all aspects related to the mathematical analysis of the Navier-Stokes equations. For other important aspects, different viewpoints and further references we refer the reader to the monographs by Constantin and Foias [22], Temam [140], von Wahl [145], Lions [77], Sohr [132], Ladhyzenskaya [65], LemariéRieusset [73] and Cannone [17], as well as to the survey (or key) articles by Leray [74], Serrin [130], Heywood [51], Galdi [45], Wiegner [146], Kozono [60], among others. Consider a (suitable) weak solution of the Navier-Stokes equations with b = 0 and with an initial condition v0 ∈ W k,2 (Ω) for all k ∈ N. Then, following also discussion in Section 5, there is certainly T ∗ > 0 such that v is a smooth flow on [0, T ∗]. Even more, such v is uniquely determined in class of weak solutions. Since 1,2 v ∈ L2 (0, ∞; Wper ) there is T ∗∗ such that v0 := v(T ∗∗ ) fulfils (B.4.40) implying that v is smooth on [T ∗∗ , ∞). Thus possible singularities lie somewhere between T ∗ and T ∗∗ . Set σ = {t ∈ h0, ∞), lim sup k∇v(τ )k2 = +∞}. τ →t 1,2 Since v ∈ L2 (0, ∞; Wper ), the Lebesgue measure of σ is zero. The program to study the structure of possible singularities was initiated by J. Leray [74], who showed that even 12 -Hausdorff dimension of σ is zero, h0, T i \ σ $LaTeX: 2005/4/20 $ 84 J. Málek and K. R. Rajagopal can be written as S∞ j=1 (aj , bj ) and if t∗ ∈ σ then kv(t)k1,2 ≥ √C t∗ −t as t → t∗− . Leray proposed to construct a weak solution exhibiting the singularity at t ∗ in the form† v(t, x) = λ(t)U(λ(t)x), p(t, x) = λ2 (t)P (λ(t)x) with λ(t) = p 2a(t∗ − t), (B.6.1) where a > 0, and showed that if there is a nontrivial solution (U, P ) of the system div U = 0, −2aU + div(y ⊗ U) + div(U ⊗ U) − ν0 4U + ∇P = 0, (B.6.2) (y is a generic point of R3 ), and if U ∈ L∞ (R3 ) ∩ L2 (R2 ), then (v, p) of the form (B.6.1) is a weak solution of the Navier-Stokes equations, being singular at t = t ∗ . Based on an observation that |U|2 2 + P + ay · U satisfies the maximum principle, Nečas, Růžička and Šverák [94] show that in the class of weak solutions satisfying U ∈ L3 (R3 ), the system (B.6.2) admits only trivial solution, U ≡ 0. Tsai [144] prove the same under more general assumptions namely if U ∈ Lq (R3 ) for q > 3 or if v fulfils energy inequality considered on any ball B ⊂ R3 . Clearly, the implication U ∈ W 1,2 (R3 ) =⇒ U ≡ 0 follows from the result established in [94]. An elementary proof of this implication is given in [79], where also so-called pseudo-selfsimilar solutions are introduced. Their nonexistence is established in [90]. Note that v of the form (B.6.1) is not only in L∞ (0, T ; L2) but also in L∞ (0, T ; L3) provided U ∈ L3 . Note also that the self-similar transformation (B.1.23) is meaningful in any conical domain. This recalls for the possibility to construct singular solution of the form (B.6.1) in cones. Escauriaza, Seregin and Šverák [30], [32], [31] show, using an approach different from that used in [94], that such solution does not exist, at least at the half-space. Consider all points (t, x) such that v is bounded (or Hölder continuous) at certain parabolic neighborhood of (t, x). Let S be the complement of such set in h0, +∞) × R3 . Scheffer [121], [122] and [123] started to study the Hausdorff dimension of the set of singularities S. Caffareli, Kohn and Nierenberg [16], introducing the notion of suitable weak solution and proving its existence, finalized these studies by showing that one-dimensional parabolic Hausdorf measure of S is zero. Simplification of the proof and certain improvements of the technique called partial regularity can be found in [75], [71] and [127], or [19]. To give a better description of the result by Caffareli, Kohn and Nierenberg, we recall the definition of (parabolic) Hausdorf measures and related statements. † The form of (v, p) can be also motivated by the self-similar scaling (B.1.23). $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations For a countable collection Q = S i∈N Bρi (yi ) in Rs , set S(α) = 85 P∞ i=1 ρα i . Then α-dimensional Hausdorf measure H α (F ) of a Borel set F ⊂ Rs is defined as H α (F ) = lim inf {S(α); F ⊂ δ→0+ Q ∞ [ Bri (xi ), sup ri < δ}. Similarly, for a countable collection Qpar = Qri (ti , xi ) = {(τ, y); |τ − ti | < i∈N i=1 ri2 , |y S Qri (ti , xi ) of parabolic balls P∞ α − xi | < ri }, set S par (α) = i=1 ri . Then i∈N α-dimensional parabolic Hausdorf measure P α (E) of a Borel set E ⊂ R × R3 is defined as P α (E) = lim inf {S par (α); E ⊂ par δ→0+ Q ∞ [ Qri (ti , xi ), sup ri < δ}. i∈N i=1 Clearly, P α (E) = 0 ⇔ ∀ ε > 0 ∃Qpar = ∞ [ Qri (ti , xi ) such that i=1 0 X riα < ε. (B.6.3) i∈N 00 If P α (E) < ∞, then P α = 0 for all α0 > α and P α (E) = +∞ for all α00 < α. If α ∈ N and P α (E) < ∞, then E is homeomorphic to a subset in Rα . The following characterization of smooth points is known due to Caffarelli, Kohn and Nirenberg [16]: Theorem 6.1. Let (v, p) be a suitable weak solution to the Navier-Stokes equations. There is a universal constant ε∗ > 0 such that if Z 1 |∇v|2 dx dt < ε∗ , R QR (t0 ,x0 ) (B.6.4) then for any k ∈ N ∪ {0}, the functions (t, x) 7→ ∇k v(t, x) are Hölder continuous in Q R (t0 , x0 ) and 2 sup |∇k v| ≤ Ck R−(k+1) , (B.6.5) (τ,y)∈Q R (t0 ,x0 ) 2 Ck being a universal constant. Thus, if (t∗ , x∗ ) is a singular point, there is QR∗ (t∗ , x∗ ) such that Z |∇v|2 dτ dx ≥ ε∗ R∗ . (B.6.6) QR∗ (t∗ ,x∗ ) Clearly, S (t∗ ,x∗ )∈S QR∗ (t∗ , x∗ ) is a collection of (parabolic) balls that cover S. Since the four-dimensional Lebesgue measure of S is zero, the four-dimensional Lebesgue measure of discovering collection can be made arbitrarily small. Vitali’s covering $LaTeX: 2005/4/20 $ 86 J. Málek and K. R. Rajagopal lemma then provides the existence of a countable subcollection of mutually disjoint balls such that S⊂ ∞ [ Q5Ri (ti , xi ) (ti , xi ) ∈ S), i=1 and the four-dimensional Lebesgue measure of say less than ε∗ ε/5, ε > 0 arbitrary. Then S∞ i=1 QRi (ti , xi ) is small as needed, Z ∞ Z 5 5 X 2 |∇v| = ∗ |∇v|2 < ε. 5Ri ≤ ∗ ε ε i ,xi ) i ,xi ),R <δ Q (t Q (t } { R R i i i i=1 i=1 ∞ X According to (B.6.3), P 1 (S) = 0 and S cannot be a curve in R+ ×R3 . Consequently, • weak solutions of the Navier-Stokes equations in two dimensions are smooth, • axially symmetric flows cannot have the singularity outside the set r = 0, • the result on zero 1 2 1 2 -dimensional 1 Hausdorff measure of singular times σ follows, due to inequality H (σ) ≤ cP (S). Schaffer in [124] constructs an irregular (non-physical) b satisfying b · v ≤ 0 so that for any δ > 0 the Hausdorff dimension of singular points is above 1−δ showing thus optimality of the Caffarelli, Kohn, Nirenberg result. We refer to the above mentioned literature for further details. 7. Other incompressible fluid models As we were invited to address both physical and analytical aspects to fluids with pressure dependent viscosities in Volume of Handbook of Mathematical Fluid Dynamics (edited by S. Friedlander and D. Serre), we only briefly comment available results here. 7.1. Fluids with pressure-dependent viscosity To our knowledge, there is no large-time and large-data existence result to system of partial differential equations of the form (B.1.3). Even more, no results on large-time existence for small data or short-time existence for large-data seems to be in place. Renardy [119] obtained local existence and uniqueness result in higher Sobolev spaces not only assuming the viscosity fulfils lim p→+∞ ν(p) = 0, p (B.7.1) $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 87 that clearly contradicts to experiments† but also requiring an additional conditions on eigenvalues of D(v) in terms of ∂ν ∂p . Gazzola does not assume (B.7.1). He however establishes only short time existence of smooth solution for small data under very restrictive conditions, both on the almost conservative specific body forces b and initial data. 7.2. Fluids with pressure and shear dependent viscosities Considering apparently a more complicated model for fluids, namely (B.1.2), where the viscosity is not only a function of p, but depends also on the shear rate, it has been observed by Málek, Nečas and Rajagopal [80] that for certain specific forms of viscosities, large-time and large-data existence takes place. More precisely, assuming that for a C 1 -function S of the form S(p, D(v)) = ν(p, |D(v)|2 )D(v) there 3×3 are two positive constants C1 , C2 such that for all 0 6= A, B ∈ Rsym and for all q∈R C1 (1 + |A|2 ) and r−2 2 |B|2 ≤ r−2 ∂S(q, A) · (B ⊗ B) ≤ C2 (1 + |A|2 ) 2 |B|2 ∂A ∂S(q, A) ≤ γ0 (1 + |A|2 ) r−2 4 , ∂q 1 C1 with γ0 = min( , ), 2 4C2 (B.7.3) (B.7.4) Málek, Nečas and Rajagopal [80] established the following result. 1,2 Theorem 7.1. Let S satisfy (B.7.3) and (B.7.4) with r ∈ ( 59 , 2). Let v0 ∈ Wper and g ∈ L2 (0, T ). Then there is a (suitable) weak solution (v, p) to (B.1.2) subjected R to spatially periodic conditions (B.2.9) and the requirement Ω p(t, x) dx = g(t) for t ∈ (0, T ) such that 5r 5r 1,r v ∈ C(0, T ; L2weak ) ∩ Lr (0, T ; Wper ) ∩ L 3 (0, T ; L 3 ) 5r 6 5r 6 p ∈ L (0, T ; L ) (B.7.5) (B.7.6) Moreover, if r ∈ h 53 , 2) there is a solution (v, p) such that 1,2 2,r v ∈ L∞ (0, T ∗ ; Wper,div ) ∩ Lr (0, T ∗ ; Wdiv ) (B.7.7) p ∈ L2 (0, T ∗ ; W 1,2 ). (B.7.8) † In fact, in most popular engineering models the relationship between ν and p is exponential, i.e., ν(p) = exp(α0 p), α0 > 0. (B.7.2) $LaTeX: 2005/4/20 $ 88 J. Málek and K. R. Rajagopal Here, T ∗ > 0 is arbitrary if v0 is sufficiently small or T ∗ is small enough if v0 is arbitrary. It may be of interest to mention that considering instead of (B.7.2) the viscosity of the form ν(p, |D(v)|2 ) = (1 + A + |D|2 ) r−2 2 (A + exp(−αqp) + |D|2 ) r−2 2 if p < 0, if p ≥ 0, the assumptions (B.7.3) and (B.7.4) are fulfilled provided that 2αq(2 − r) ≤ (r − 1)A 2−r 2 . This can be achieved by taking one of the parameters α or q small enough, or A large enough or r close enough to 2. Another examples and the proof of Theorem 7.1 can be found in [80]. Twodimensional flows are studied in [53] and [13]. In the latter, large-time behavior, based on uniqueness result, is also studied via the method of trajectories. A step towards the treatment of other boundary conditions is made in [41]. 7.3. Inhomogeneous incompressible fluids Here, we give references to results relevant to analysis of the partial differential equations (B.1.1). The first results deal with T of the form T = −pI + 2µ(ρ)D(v). Large-time and large-data existence of weak solution established by Novosibirsk school prior 1990 is presented by Antontsev, Kazhikov and Monakhov in [6]. A profound exposition is given in the first chapter of the monograph by P. L. Lions [77]. The fluids with µ depending on |D(v)|2 were analysed in in [36], were FernadézCara, Guillén and Ortega proved existence of weak solution to (B.1.1) with for r ≥ 12 5 . T = −I + µ0 + µ1 |D(v)|r−2 D(v) This result, treating homogeneous Dirichlet, i.e., (no-slip) boundary conditions, was recently improved by Guillén-Gonzáles [49], in the case of spatially periodic problem to r ≥ 2. $LaTeX: 2005/4/20 $ Some generalizations of the Navier-Stokes equations 89 Bibliography [1] The Oxford English Dictionary. The Oxford University Press, 1989. [2] H. 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