Mathematical theory of compressible viscous fluids Eduard Feireisl and Milan Pokorný 2 Another advantage of a mathematical statement is that it is so definite that it might be definitely wrong. . . Some verbal statements have not this merit. F.L.Richardson (1881-1953) Contents I Mathematical fluid dynamics of compressible fluids 5 1 Introduction 7 2 Mathematical model 2.1 Mass conservation . . . . . . . . . . . . . 2.2 Balance of momentum . . . . . . . . . . 2.3 Spatial domain and boundary conditions 2.4 Initial conditions . . . . . . . . . . . . . . . . . 9 9 10 12 13 . . . . . 15 15 16 16 17 18 4 A priori bounds 4.1 Total mass conservation . . . . . . . . . . . . . . . . . . . . . 4.2 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Pressure estimates . . . . . . . . . . . . . . . . . . . . . . . . 21 21 22 24 . . . . 3 Weak solutions 3.1 Equation of continuity – weak formulation 3.1.1 Weak-strong compatibility . . . . . 3.1.2 Weak continuity . . . . . . . . . . . 3.1.3 Total mass conservation . . . . . . 3.2 Balance of momentum - weak formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Complete weak formulation 27 5.1 Equation of continuity . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Momentum equation . . . . . . . . . . . . . . . . . . . . . . . 28 5.3 Energy inequality . . . . . . . . . . . . . . . . . . . . . . . . . 29 3 4 II CONTENTS Weak sequential stability for large γ 6 Weak sequential stability 6.1 Uniform bounds . . . . . . . . . . . . . . 6.2 Limit passage . . . . . . . . . . . . . . . 6.3 Compactness of the convective term . . . 6.3.1 Compactness via Div-Curl lemma 6.4 Passing to the limit - step 1 . . . . . . . 6.5 Strong convergence of the densities . . . 6.5.1 Renormalized equation . . . . . . 6.5.2 The effective viscous flux . . . . . III . . . . . . . . . . . . . . . . . . . . . . . . 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Existence of weak solutions for small γ 33 34 35 36 36 38 39 39 42 49 7 Mathematical tools 53 7.1 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . 53 7.2 Continuity in time . . . . . . . . . . . . . . . . . . . . . . . . 62 8 Existence proof 8.1 Approximations . . . . . . . . . . . . . . . . . . . 8.2 Existence for the Galerkin approximation . . . . . 8.3 Estimates independent of n, limit passage n → ∞ 8.4 Estimates independent of ε, limit passage ε → 0+ 8.4.1 Strong convergence of the density . . . . . 8.5 Estimates independent of δ, limit passage δ → 0+ 8.5.1 Strong convergence of the density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 69 70 80 84 89 91 96 Part I Mathematical fluid dynamics of compressible fluids 5 Chapter 1 Introduction Despite the concerted effort of generations of excellent mathematicians, the fundamental problems in partial differential equations related to continuum fluid mechanics remain largely open. Solvability of the Navier-Stokes system describing the motion of an incompressible viscous fluid is one in the sample of millenium problems proposed by Clay Institute, see [7]. In contrast with these apparent theoretical difficulties, the Navier-Stokes system became a well established model serving as a reliable basis of investigation in continuum fluid mechanics, including the problems involving turbulence phenomena. An alternative approach to problems in fluid mechanics is based on the concept of weak solutions. As a matter of fact, the balance laws, expressed in classical fluid mechanics in the form of partial differential equations, have their origin in integral identities that seem to be much closer to the modern weak formulation of these problems. Leray [11] constructed the weak solutions to the incompressible Navier-Stokes system as early as in 1930, and his “turbulent solutions” are still the only ones available for investigating large data and/or problems on large time intervals. Recently, the real breakthrough is the work of Lions [12] who generalized Leray’s theory to the case of barotropic compressible viscous fluids (see also Vaigant and Kazhikhov [17]). The quantities playing a crucial role in the description of density oscillations as the effective viscous flux were identified and used in combination with a renormalized version of the equation of continuity to obtain first large data/large time existence results in the framework of compressible viscous fluids. The main goal of the this lecture series is to present the mathematical theory of compressible barotropic fluids in the framework of Lions [12], to7 8 CHAPTER 1. INTRODUCTION gether with the extensions developed in [8]. After an introductory part we first focus on the crucial question of stability of a family of weak solutions that is the core of the abstract theory, with implications to numerical analysis and the associated real world applications. For the sake of clarity of presentation, we discuss first the case, where the pressure term has sufficient growth for large value of the density yielding sufficiently strong energy bounds. We also start with the simplest geometry of the physical space, here represented by a cube, on the boundary of which the fluid satisfies the slip boundary conditions. As is well-known, such a situation may be reduced to studying the purely spatially periodic case, where the additional difficulties connected with the presence of boundary conditions is entirely eliminated. Next part of this lecture series will be devoted to the detailed existence proof with (nowadays) optimal restriction on the pressure function. We will also consider the case of homogeneous Dirichlet boundary conditions. Chapter 2 Mathematical model As the main goal of this lecture series is the mathematical theory, we avoid a detailed derivation of the mathematical model of a compressible viscous fluid. Remaining on the platform of continuum fluid mechanics, we suppose that the motion of a compressible barotropic fluid is described by means of two basic fields: the mass density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ̺ = ̺(t, x), the velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u = u(t, x), functions of the time t ∈ R and the spatial position x ∈ R3 . 2.1 Mass conservation Let us recall the classical argument leading to the mathematical formulation of the physical principle of mass conservation, see e.g. Chorin and Marsden [3]. Consider a volume B ⊂ R3 containing a fluid of density ̺. The change of the total mass of the fluid contained in B during a time interval [t1 , t2 ], t1 < t2 is given as Z Z ̺(t2 , x) dx − ̺(t1 , x) dx. B B One of the basic laws of physics incorporated in continuum mechanics as the principle of mass conservation asserts that mass is neither created nor destroyed. Accordingly, the change of the fluid mass in B is only because of the mass flux through the boundary ∂B, here represented by ̺u · n, where 9 10 CHAPTER 2. MATHEMATICAL MODEL n denotes the outer normal vector to ∂B: Z Z Z t2 Z ̺(t2 , x) dx − ̺(t1 , x) dx = − ̺(t, x)u(t, x) · n(x) dSx dt. (2.1) B B t1 ∂B One should remember formula (2.1) since it contains all relevant piece of information provided by physics. The following discussion is based on mathematical arguments based on the (unjustified) hypotheses of smoothness of all field in question. To begin, apply Gauss-Green theorem to rewrite (2.1) in the form: Z Z Z t2 Z ̺(t2 , x) dx − ̺(t1 , x) dx = − divx ̺(t, x)u(t, x) dx dt. B B t1 B Furthermore, fixing t1 = t and performing the limit t2 → t we may use the mean value theorem to obtain Z Z Z 1 ∂t ̺(t, x) dx = lim ̺(t2 , x) dx − ̺(t, x) dx (2.2) t2 →t t2 − t B B B Z t2 Z 1 divx ̺(t, x)u(t, x) dx dt = − lim t2 →t t2 − t t B Z = − divx ̺(t, x)u(t, x) dx. B Finally, as relation (2.2) should hold for any volume element B, we may infer that ∂t ̺(t, x) + divx ̺(t, x)u(t, x) = 0. (2.3) Relation (2.3) is a first order partial differential equation called equation of continuity. 2.2 Balance of momentum Using arguments similar to the preceding part, we derive balance of momentum in the form ∂t ̺(t, x)u(t, x) +divx ̺(t, x)u(t, x)⊗u(t, x) = divx T(t, x)+̺(t, x)f (t, x), (2.4) 11 2.2. BALANCE OF MOMENTUM or, equivalently (cf. (2.3)), h i ̺(t, x) ∂t u(t, x) + u(t, x) · ∇x u(t, x) = divx T(t, x) + ̺(t, x)f (t, x), where the tensor T is the Cauchy stress and f denotes the (specific) external force acting on the fluid. We adopt the standard mathematical definition of fluids in the form of Stokes’ law T = S − pI, where S is the viscous stress and p is a scalar function termed pressure. In addition, we suppose that the viscous stress is a linear function of the velocity gradient, specifically S obeys Newton’s rheological law 2 t S = S(∇x u) = µ ∇x u + ∇x u − divx uI + ηdivx uI, (2.5) 3 with the shear viscosity coefficient µ and the bulk viscosity coefficient η, here assumed constant, µ > 0, η ≥ 0. In order to close the system, we suppose the fluid is barotropic, meaning the pressure p is an explicitly given function of the density p = p(̺). Accordingly, 2 divx T = µ∆u + (λ + µ)∇x divx u − ∇x p(̺), µ > 0, λ ≥ − µ, 3 and equations (2.3), (2.4) can be written in a concise form as Navier-Stokes system ∂t ̺ + divx (̺u) = 0, (2.6) ∂t (̺u) + divx (̺u ⊗ u) + ∇x p(̺) = µ∆u + (λ + µ)∇x divx u + ̺f . (2.7) The system of equations (2.6), (2.7) should be compared with a “more famous” incompressible Navier-Stokes system, where the density is constant, say ̺ ≡ 1, while (2.6), (2.7) “reduces” to divx u = 0, (2.8) 12 CHAPTER 2. MATHEMATICAL MODEL ∂t u + divx (u ⊗ u) + ∇x p = µ∆u + f . (2.9) Unlike in (2.7), the pressure p in (2.9) is an unknown function determined (implicitly) by the fluid motion! The pressure in the incompressible NavierStokes system has non-local character and may depend on the far field behavior of the fluid system. 2.3 Spatial domain and boundary conditions In the real world applications, the fluid is confined to a bounded spatial domain Ω ⊂ R3 . The presence of the physical boundary ∂Ω and the associated problem of fluid-structure interaction represent a source of substantial difficulties in the mathematical analysis of fluids in motion. In order to avoid technicalities, we suppose in Part II of the lecture series that the motion is space-periodic, specifically, ̺(t, x) = ̺(t, x + ai ), u(t, x) = u(t, x + ai ) for all t, x, (2.10) where the period vectors a1 = (a1 , 0, 0), a2 = (0, a2 , 0), a3 = (0, 0, a3 ) are given. Equivalently, we may assume that Ω is a flat torus, Ω = [0, a1 ]|{0,a1 } × [0, a2 ]|{0,a2 } × [0, a3 ]|{0,a3 } . The space-periodic boundary conditions have a nice physical interpretation in fluid mechanics, see Ebin [5]. Indeed, if we restrict ourselves to the classes of functions defined on the torus Ω and satisfying the extra geometric restrictions: ̺(t, x) = ̺(t, −x), ui (t, ·, xi , ·) = −ui (t, ·, −xi , ·), i = 1, 2, 3, ui (t, ·, xj , ·) = ui (t, ·, −xj , ·) for i 6= j, and, similarly, fi (t, ·, xi , ·) = −fi (t, ·, −xi , ·), fi (t, ·, xj , ·) = fi (t, ·, −xj , ·) for i 6= j, we can check that • the equations (2.6), (2.7) are invariant with respect to the above transformations; 13 2.4. INITIAL CONDITIONS • the velocity field u satisfies the so-called complete slip conditions u · n = 0, [Sn] × n = 0 (2.11) on the boundary of the spatial block [0, a1 ] × [0, a2 ] × [0, a3 ]. We remark that the most commonly used boundary conditions for viscous fluids confined to a general spatial domain Ω (not necessarily a flat torus) are the no-slip u|∂Ω = 0. (2.12) We will focus on this type of the boundary condition in Part III of this lecture series. As a matter of fact, the problem of the choice of correct boundary conditions in the real world applications is rather complex, some parts of the boundaries may consist of a different fluid in motion, or the fluid domain is not a priori known (free boundary problems). The interested reader may consult Priezjev and Troian [15] for relevant discussion. 2.4 Initial conditions Given the initial state at a reference time t0 , say t0 = 0, the time evolution of the fluid is determined as a solution of the Navier-Stokes system (2.6), (2.7). It is convenient to introduce the initial density ̺(0, x) = ̺0 (x), x ∈ Ω, (2.13) together with the initial distribution of the momentum, (̺u)(0, x) = (̺u)0 (x), x ∈ Ω, (2.14) as, strictly speaking, the momentum balance (2.7) is an evolutionary equation for ̺u rather than u. Such a difference will become clear in the so-called weak formulation of the problem discussed in the forthcoming section. 14 CHAPTER 2. MATHEMATICAL MODEL Chapter 3 Weak solutions A vast class of non-linear evolutionary problems arising in mathematical fluid mechanics is not known to admit classical (differentiable, smooth) solutions for all choices of data and on an arbitrary time interval. On the other hand, most of the real world problems call for solutions defined in-the-large approached in the numerical simulations. In order to perform a rigorous analysis, we have to introduce a concept of generalized or weak solutions, for which derivatives are interpreted in the sense of distributions. The dissipation represented by viscosity should provide a strong regularizing effect. Another motivation, at least in the case of the compressible Navier-Stokes system (2.6), (2.7), is the possibility to study the fluid dynamics emanating from irregular initial state, for instance, the density ̺0 may not be continuous. As shown by Hoff [9], the singularities incorporated initially will “survive” in the system at any time; thus the weak solutions are necessary in order to describe the dynamics. 3.1 Equation of continuity – weak formulation We consider equation (2.6) on the space-time cylinder (0, T ) × Ω, where Ω is the flat torus introduced in Section 2.3. Multiplying (2.6) on ϕ ∈ Cc∞ ((0, T ) × Ω), integrating the resulting expression over (0, T ) × Ω, and performing by-parts integration, we obtain Z TZ ̺(t, x)∂t ϕ(t, x) + ̺(t, x)u(t, x) · ∇x ϕ(t, x) dx dt = 0. (3.1) 0 Ω 15 16 CHAPTER 3. WEAK SOLUTIONS Definition 3.1 We say that a pair of functions ̺, u is a weak solution to equation (2.6) in the space-time cylinder (0, T ) × Ω if ̺, ̺u are locally integrable in (0, T ) × Ω and the integral identity (3.1) holds for any test function ϕ ∈ Cc∞ ((0, T ) × Ω). 3.1.1 Weak-strong compatibility It is easy to see that any classical (smooth) solution of equation (2.6) is also a weak solution. Similarly, any weak solution that is continuously differentiable satisfies (2.6) pointwise. Such a property is called weak-strong compatibility. 3.1.2 Weak continuity Up to now, we have left apart the problem of satisfaction of the initial condition (2.13). Obviously, some kind of weak continuity is needed for (2.13) to make sense. To this end, we make an extra hypothesis, namely, ̺u ∈ L1 (0, T ; L1 (Ω; R3 )). (3.2) Taking ϕ(t, x) = ψ(t)φ(x), ψ ∈ Cc∞ (0, T ), φ ∈ Cc∞ (Ω) as a test function in (3.1) we may infer, by virtue of (3.2), that the function Z t 7→ ̺(t, x)φ(x) dx is absolutely continuous in [0, T ] (3.3) Ω for any φ ∈ Cc∞ (Ω). In particular, the initial condition (2.13) may be satisfied in the sense that Z Z lim ̺(t, x)φ(x) dx = ̺0 (x)φ(x) dx for any φ ∈ Cc∞ (Ω). t→0+ Ω Ω Now, take ϕε (t, x) = ψε (t)ϕ(t, x), ϕ ∈ Cc∞ ([0, T ] × Ω), 3.1. EQUATION OF CONTINUITY – WEAK FORMULATION 17 where ψε ∈ Cc∞ (0, τ ), 0 ≤ ψε ≤ 1, ψε ր 1[0,τ ] as ε → 0. Taking ϕε as a test function in (3.1) and letting ε → 0, we conclude, making use of (3.3), that Z Z ̺(τ, x)ϕ(τ, x) dx − ̺0 (x)ϕ(0, x) dx (3.4) = Z Ω τ 0 Z Ω Ω ̺(t, x)∂t ϕ(t, x) + ̺(t, x)u(t, x) · ∇x ϕ(t, x) dx dt for any τ ∈ [0, T ] and any ϕ ∈ Cc∞ ([0, T ] × Ω). Formula (3.4) can be alternatively used a definition of weak solution to problem (2.6), (2.13). It is interesting to compare (3.4) with the original integral formulation of the principle of mass conservation stated in (2.1). To this end, we take ϕε (t, x) = φε (x), with φε ∈ Cc∞ (B) such that 0 ≤ φε ≤ 1, φε ր 1B as ε → 0. Z It is easy to see that Ω ̺(τ, x)ϕε (τ, x) dx − Z Ω ̺0 (x)ϕε (0, x) dx → Z B ̺(τ, x) dx − Z ̺0 (x) dx B as ε → 0, which coincides with the expression on the left-hand side of (2.1). Consequently, the right-hand side of (3.4) must posses a limit and we set Z τZ Z τZ ̺(t, x)u(t, x) · n dSx dt. ̺(t, x)u(t, x) · ∇x φε (x) dx dt → − 0 0 Ω ∂B In other words, the weak solutions possess a normal trace on the boundary of the cylinder (0, τ ) × B that satisfies (2.1), see Chen and Frid [2] for more elaborate treatment of the normal traces of solutions to conservation laws. 3.1.3 Total mass conservation Taking ϕ = 1 for t ∈ [0, τ ] in (3.4) we obtain Z Z ̺(τ, x) dx = ̺0 (x) dx = M0 for any τ ≥ 0, Ω Ω meaning, the total mass M0 of the fluid is a constant of motion. (3.5) 18 3.2 CHAPTER 3. WEAK SOLUTIONS Balance of momentum - weak formulation Similarly to the preceding part, we introduce a weak formulation of the balance of momentum (2.7): Definition 3.2 The functions ̺, u represent a weak solution to the momentum equation (2.7) in the set (0, T ) × Ω if the integral identity Z T 0 Z (̺u)(t, x)∂tϕ (t, x) + (̺u ⊗ u)(t, x) : ∇xϕ (t, x) (3.6) Ω +p(̺)(t, x)divxϕ (t, x) dx dt Z TZ µ∇x u(t, x) : ∇xϕ (t, x) = 0 Ω +(λ + µ)divx u(t, x)divxϕ (t, x) − ̺(t, x)f (t, x) · ϕ (t, x) dx dt is satisfied for any test function ϕ ∈ Cc∞ ((0, T ) × Ω; R3 ). Of course, we have tacitly assumed that all quantities appearing in (3.6) are at least locally integrable in (0, T ) × Ω. In particular, as (3.6) contains explicitly ∇x u, we have to assume integrability of this term. As we shall see in the following section, one can expect, given the available a priori bounds, ∇x u to be square integrable, specifically, u ∈ L2 (0, T ; W 1,2 (Ω; R3 )). If Ω ⊂ R3 is a (bounded) domain with a non-void boundary, we can enforce several kinds of boundary conditions by means of the properties of the test functions. Thus, for instance, the no-slip boundary conditions u|∂Ω = 0, (3.7) require the integral identity (3.6) to be satisfied for any compactly supported 3.2. BALANCE OF MOMENTUM - WEAK FORMULATION 19 test function ϕ, while u ∈ L2 (0, T ; W01,2 (Ω; R3 )), where W01,2 (Ω; R3 ) is the Sobolev space obtained as the closure of Cc∞ (Ω; R3 ) in the W 1,2 -norm. Remark 3.1 We may get the weak-strong compatibility as in the case of continuity equation. 20 CHAPTER 3. WEAK SOLUTIONS Chapter 4 A priori bounds A priori bounds are natural constraints imposed on the set of (hypothetical) smooth solutions by the data as well as by the differential equations satisfied. A priori bounds determine the function spaces framework the (weak) solutions are looked for. By definition, they are formal, derived under the principal hypothesis of smoothness of all quantities in question. 4.1 Total mass conservation . The fluid density ̺ satisfies the equation of continuity that may be written in the form ∂t ̺ + u · ∇x ̺ = −̺divx u. This is a transport equation with the characteristic field defined d X(t, x0 ) = u(t, X), X(0, x0 ) = x0 . dt Accordingly, (4.1) can be written as d ̺(t, X(t, ·)) = −̺(t, X(t, ·))divx u(t, X(t, ·)). dt Consequently, we obtain 21 (4.1) 22 CHAPTER 4. A PRIORI BOUNDS inf ̺(0, x) exp −tkdivx ukL∞ ((0,T )×Ω) x∈Ω ≤ ̺(t, x) ≤ ≤ sup ̺(0, x) exp tkdivx ukL∞ ((0,T )×Ω) x∈Ω for any t ∈ [0, T ]. (4.2) Unfortunately, the bounds established in (4.2) depend on kdivx ukL∞ on which we have no information. Thus we may infer only that ̺(t, x) ≥ 0. (4.3) Relation (4.3) combined with the total mass conservation (3.5) yield k̺(t, ·)kL1 (Ω) = k̺0 kL1 (Ω) , ̺(0, ·) = ̺0 . 4.2 (4.4) Energy balance Taking the scalar product of the momentum equation (2.4) with u we deduce the kinetic energy balance equation 1 1 2 2 ̺|u| +divx ̺|u| u +divx (p(̺)u)−p(̺)divx u−divx (Su)+S : ∇x u ∂t 2 2 (4.5) = ̺f · u. Our goal is to integrate (4.5) by parts in order to deduce a priori bounds. Imposing the no-slip boundary condition (2.12) or the space-periodic boundary condition (2.10) we get Z Z Z Z d 1 2 dx − p(̺)divx u dx + S : ∇x u dx = ̺f · u dx, ̺|u| dt Ω 2 Ω Ω Ω where, in accordance with (2.7), S : ∇x u = µ|∇x u|2 + 3(λ + µ)|divx u|2 ≥ c|∇x u|2 , c > 0, provided λ + 2/3µ > 0. (4.6) 23 4.2. ENERGY BALANCE Seeing that Z Z Z Z 1 √ √ 2 ̺f · u dx ≤ |f | ̺ ̺|u| dx ≤ kf kL∞ ((0,T )×Ω) ̺ dx + ̺|u| dx 2 Ω Ω Ω Ω we focus on the integral Z p(̺)divx u dx. Ω Multiplying the equation of continuity (4.1) by b′ (̺) we obtain the renormalized equation of continuity ∂t b(̺) + divx (b(̺)u) + b′ (̺)̺ − b(̺) divx u = 0. (4.7) Consequently, in particular, the choice b(̺) = P (̺) ≡ ̺ Z ̺ 1 p(z) dz z2 leads to b′ (̺)̺ − b(̺) = p(̺). Thus Z Z d − p(̺)divx u dx = P (̺) dx, dt Ω Ω and we deduce the total energy balance Z Z Z 1 d 2 ̺|u| + P (̺) dx + S : ∇x u dx = ̺f · u dx. dt Ω 2 Ω Ω (4.8) We conclude with Energy estimates: √ sup k ̺u(t, ·)kL2 (Ω;R3 ) ≤ c(E0 , T ), t∈[0,T ] sup t∈[0,T ] Z Z Ω P (̺)(t, ·) dx ≤ c(E0 , T ), (4.9) (4.10) T 0 ku(t, ·)k2W 1,2 (Ω;R3 ) dt ≤ c(E0 , T ), (4.11) 24 CHAPTER 4. A PRIORI BOUNDS where E0 denotes the initial energy Z 1 2 ̺0 |u0 | + P (̺0 ) dx. E0 = 2 Ω 4.3 Pressure estimates A seemingly direct way to pressure estimates is to “compute” the pressure in the momentum balance (2.7): p(̺) = −∆−1 divx ∂t (̺u)−∆−1 divx divx (̺u⊗u)+∆−1 divx divx S+∆−1 divx (̺f ), where ∆−1 is an “inverse” of the Laplacean. In order to justify this formal step, we use the so-called Bogovskii operator B ≈ div−1 x . We multiply equation (2.7) on Z 1 B[̺] = B b(̺) − b(̺) dx |Ω| Ω and integrate by parts to obtain Z TZ 0 1 = |Ω| Z − Z T 0 T 0 Z Z Ω p(̺) dx Ω Z p(̺)b(̺) dx dt b(̺) dx dt + Ω ̺u ⊗ u : ∇x B[̺] dx − Z Ω (4.12) Ω Z Z T 0 T 0 Z Ω Z Ω S : ∇x B[̺] dx dt ̺u · ∂t B[̺] dx dt (̺u · B[̺](τ, ·) − ̺0 u0 · B[̺0 ]) dx. Furthermore, we have ′ ∂t B[̺] = −B divx (b(̺)u) + b (̺)̺ − b(̺) divx u 1 − |Ω| h Z ′ b (̺)̺ − b(̺) divx u dx . Ω We recall the basic properties of the Bogovskii operator: (4.13) 25 4.3. PRESSURE ESTIMATES Bogovskii operator: p divx B[h] = h for any h ∈ L (Ω), Z Ω h dx = 0, 1 < p < ∞, B[h]|∂Ω = 0. (4.14) kB[h]kW 1,p (Ω;R3 ) ≤ c(p)khkLp (Ω) , 1 < p < ∞, (4.15) kB[h]kLq (Ω) ≤ kgkLq (Ω;R3 ) (4.16) 0 for h ∈ Lp (Ω), h = divx g, g · n|∂Ω = 0, 1 < q < ∞. As will be seen in the last part (Chapter 8), we can show that for b(̺) = ̺θ the right-hand side is possible to estimate provided nγ 2 o θ ≤ min , γ−1 . (4.17) 2 3 Note that for γ ≤ 6 the restriction comes from the second term in (4.17) while for γ > 6 the first term is more restrictive. 26 CHAPTER 4. A PRIORI BOUNDS Chapter 5 Complete weak formulation A complete weak formulation of the (compressible) Navier-Stokes system takes into account both the renormalized equation of continuity and the energy inequality. Here and hereafter we assume that Ω ⊂ R3 is either a bounded domain with Lipschitz boundary or a periodic box. For the sake of definiteness, we take the pressure in the form p(̺) = a̺γ , with a > 0 and γ > 3/2. (5.1) In Part II we restrict ourselves to the case when γ is “sufficiently” large, Part III will contain the proof only under restriction (5.1). 5.1 Equation of continuity Let us introduce a class of (nonlinear) functions b such that b ∈ C 1 [0, ∞), b(0) = 0, b′ (r) = 0 whenever r ≥ Mb . (5.2) We say that ̺, u is a (renormalized) solution of the equation of continuity (2.3), supplemented with the initial condition, ̺(0, ·) = ̺0 , if ̺ ∈ Cweak ([0, T ]; Lγ (Ω)), ̺ ≥ 0, u ∈ L2 (0, T ; W 1,2 (Ω; R3 )), and the integral identity 27 28 CHAPTER 5. COMPLETE WEAK FORMULATION Z T 0 Z Ω ′ (̺ + b(̺)) ∂t ϕ+(̺ + b(̺)) u·∇x ϕ+(b(̺) − b (̺)̺) divx uϕ dx dt =− Z Ω (5.3) (̺0 + b(̺0 )) ϕ(0, ·) dx is satisfied for any ϕ ∈ Cc∞ ([0, T ) × Ω) and any b belonging to the class specified in (5.2). In particular, taking b ≡ 0 we deduce the standard weak formulation of (2.3) in the form Z Ω = Z τ 0 ̺(τ, ·)ϕ(τ, ·) − ̺0 ϕ(0, ·) dx Z Ω (5.4) ̺∂t ϕ + ̺u · ∇x ϕ dx dt for any τ ∈ [0, T ] and any ϕ ∈ Cc∞ ([0, T ] × Ω). In case of space-periodic boundary conditions we assume ̺ and u spaceperiodic, while for the homogeneous Dirichlet boundary conditions we assume u ∈ L2 (0, T ; W01,2 (R3 ; R3 )). Note that (5.4) actually holds on the whole physical space R3 provided (in case of the Dirichlet boundary conditions) ̺, u were extended to be zero outside Ω. Note also that (5.4) implies that the initial condition ̺(0, ·) = ̺0 (·) is fulfilled. In case of the space periodic boundary conditions we may extend the functions outside Ω due to the periodicity. 5.2 Momentum equation In addition to the previous assumptions we suppose that ̺u ∈ Cweak ([0, T ]; Lq (Ω; R3 )) for a certain q > 1, p(̺) ∈ L1 ((0, T ) × Ω). The weak formulation of the momentum equation reads: 29 5.3. ENERGY INEQUALITY = − Z Z τ 0 Z ̺u(τ, ·) · ϕ (τ, ·) − (̺u)0 · ϕ (0, ·) dx (5.5) Ω τ 0 Z ̺u · ∂tϕ + ̺u ⊗ u : ∇xϕ + p(̺)divxϕ dx dt Ω Z Ω µ∇x u : ∇xϕ + (λ + µ)divx udivxϕ − ̺f · ϕ dx dt for any τ ∈ [0, T ] and for any test function ϕ ∈ Cc∞ ([0, T ] × Ω; R3 ). Note that (5.5) already includes the satisfaction of the initial condition ̺u(0, ·) = (̺u)0 . 5.3 Energy inequality The weak solutions are not known to be uniquely determined by the initial data. Therefore it is desirable to introduce as much physically grounded conditions as allowed by the construction of the weak solutions. One of them is Energy inequality: Z Ω Z τZ 1 2 µ|∇x u|2 + (λ + µ)|divx u|2 dx dt ̺|u| + P (̺) (τ, ·) dx + 2 0 Ω (5.6) Z τZ Z 1 ̺f · u dx dt |(̺u)0 |2 + P (̺0 ) dx + ≤ 2̺ 0 0 Ω Ω for a.a. τ ∈ (0, T ), where P (̺) = ̺ Z ̺ 1 p(z) dz. z2 30 CHAPTER 5. COMPLETE WEAK FORMULATION Some remarks are in order. To begin, given the specific choice of the pressure p(̺) = a̺γ and the fact that the total mass of the fluid is a constant of motion, the function P (̺) in (5.6) can be taken as P (̺) = a γ ̺ . γ−1 Next, we need a kind of compatibility condition between ̺0 and (̺u)0 provided we allow the initial density to vanish on a nonempty set: (̺u)0 = 0 a.a. on the “vacuum” set {x ∈ Ω | ̺0 (x) = 0}. (5.7) Part II Weak sequential stability for large γ 31 Chapter 6 Weak sequential stability The problem of weak sequential stability may be stated as follows: Weak sequential stability: Given a family {̺ε , uε }ε>0 of weak solutions of the compressible NavierStokes system, emanating from the initial data ̺ε (0, ·) = ̺0,ε , (̺u)(0, ·) = (̺u)0,ε , we want to show that ̺ε → ̺, uε → u as ε → 0 in a certain sense and at least for suitable subsequences, where ̺, u is another weak solution of the same system. Although showing weak sequential stability does not provide an explicit proof of existence of the weak solutions, its verification represents one of the prominent steps towards a rigorous existence theory for a given system of equations. 33 34 CHAPTER 6. WEAK SEQUENTIAL STABILITY 6.1 Uniform bounds To begin the analysis, we need uniform bounds in terms of the data. To this end, we choose the initial data in such a way that Z 1 2 |(̺u)0,ε | + P (̺0,ε ) dx ≤ E0 , (6.1) 2̺0,ε Ω where the constant E0 is independent of ε. Moreover, the main and most difficult steps of the proof of weak sequential stability remain basically the same under the simplifying assumption f ≡ 0. In accordance with the energy inequality (5.6), we get sup k̺ε (t, ·)kLγ (Ω) ≤ c (6.2) √ ess sup k ̺ε uε (t, ·)kL2 (Ω;R3 )) ≤ c, (6.3) t∈(0,T ) and t∈(0,T ) together with Z T 0 kuε (t, ·)k2W 1,2 (Ω;R3 ) dt ≤ c, (6.4) where the symbol c stands for a generic constant independent of ε. Interpolating (6.2), (6.3), we get √ √ √ √ k̺ε uε kLq (Ω;R3 ) = k ̺ε ̺ε uε kLq (Ω;R3 ) ≤ k ̺ε kL2γ (Ω) k ̺ε uε kL2 (Ω;R3 ) , with q= 2γ > 1 provided γ > 1. γ+1 We conclude that supt∈[0,T ] k̺ε uε (t, ·)kLq (Ω;R3 ) , q = 2γ . γ+1 (6.5) Next, applying a similar treatment to the convective term in the momentum equation, we have k̺ε uε ⊗ uε kLq (Ω;R3×3 ) = k̺ε uε kL2γ/(γ+1) (Ω;R3 ) kuε kL6 (Ω;R3 ) , with q = 6γ . 4γ + 3 35 6.2. LIMIT PASSAGE Using the standard embedding relation W 1,2 (Ω) ֒→ L6 (Ω), (6.6) we may therefore conclude that Z T 0 k̺ε uε ⊗ uε k2Lq (Ω;R3×3 ) dt ≤ c, q = Note that 6γ . 4γ + 3 (6.7) 6γ 3 > 1 as long as γ > . 4γ + 3 2 Finally, we have the pressure estimates (see Chapter 8 for the proof): Z T 0 Z Ω p(̺ε )̺αε dx dt = a Z T 0 Z Ω ̺γ+α dx dt ≤ c ε (6.8) for α = min{ γ2 , 23 γ − 1}. 6.2 Limit passage In view of the uniform bounds established in the previous section, we may assume that ̺ε → ̺ weakly-(*) in L∞ (0, T ; Lγ (Ω)), (6.9) uε → u weakly in L2 (0, T ; W 1,2 (Ω; R3 )) (6.10) passing to suitable subsequences as the case may be. Moreover, since ̺ε satisfies the equation of continuity (5.4), (6.9) may be strengthened to (see Chapter 7) ̺ε → ̺ in Cweak ([0, T ]; Lγ (Ω)). (6.11) Let us recall that, in view of (6.9), relation (6.11) simply means Z Z n o n o t 7→ ̺ε (t, ·)ϕ dx → t 7→ ̺(t, ·)ϕ dx in C[0, T ] Ω for any ϕ ∈ Cc∞ (Ω). Ω 36 6.3 CHAPTER 6. WEAK SEQUENTIAL STABILITY Compactness of the convective term Our next goal is to establish convergence of the convective terms. Recall that, in view of the estimate (6.5), we may suppose that ̺ε uε → ̺u weakly-(*) in L∞ (0, T ; L2γ/(γ+1) (Ω; R3 )) and even ̺ε uε → ̺u in Cweak ([0, T ] : L2γ/(γ+1) (Ω; R3 )), (6.12) where the bar denotes (and will always denote in the future) a weak limit of a composition. Our goal is to show that ̺u = ̺u. This can be observed in several ways. Seeing that W01,2 (Ω) ֒→֒→ Lq (Ω) compactly for 1 ≤ q < 6, we deduce that 6 Lp (Ω) ֒→֒→ W −1,2 (Ω) compactly whenever p > . 5 (6.13) In particular, relation (6.12) yields ̺ε uε → ̺u in C([0, T ]; W −1,2 (Ω))), which, combined with (6.10), gives rise to the desired conclusion ̺u = ̺u. For more details see again Chapter 7. 6.3.1 Compactness via Div-Curl lemma Div-Curl lemma, developed by Murat and Tartar [13], [16], represents an efficient tool for handling compactness in non-linear problems, where the classical Rellich-Kondraschev argument is not applicable. 6.3. COMPACTNESS OF THE CONVECTIVE TERM 37 Div-Curl lemma: Lemma 6.1 Let B ⊂ RM be an open set. Suppose that vn → v weakly in Lp (B; RM ), wn → w weakly in Lq (B; RM ) as n → ∞, where 1 1 1 + = < 1. p q r Let, moreover, −1,s {div[v]}∞ (B), n=1 be precompact in W −1,s {curl[w]}∞ (B, RM ×M ) n=1 be precompact in W for a certain s > 1. Then vn · wn → v · w weakly in Lr (B). We give the proof only for a very special case that will be needed in the future, namely, we assume that Z Φn dy = 0. (6.14) div vn = 0, wn = ∇x Φn , RM Moreover, given the local character of the weak convergence, it is enough to show the result for B = RM . By the same token, we may assume that all functions are compactly supported. We recall that a (scalar) sequence −1,s {gn }∞ (RM ) if n=1 is precompact in W M M s gn = div[hn ], with {hn }∞ n=1 precompact in L (R ; R ). Now, it follows from the standard compactness arguments that Φn → Φ (strongly) in Lq (RM ), ∇x Φ = w. 38 CHAPTER 6. WEAK SEQUENTIAL STABILITY Taking ϕ ∈ Cc∞ (RM ) we have Z Z vn · wn ϕ dy = RM Z =− RM RM vn · ∇x Φn ϕ dy vn · ∇x ϕΦn dy → − = Z RM Z RM v · ∇x ϕΦ dy v · wϕ dy, which completes the proof under the simplifying hypothesis (6.14). Now, compactness of the product term ̺ε uε can be viewed by a direct application of Div-Curl lemma in the space-time R4 , with the choice vε = [̺ε , ̺ε uε ], wε = [uj,ε , 0, 0, 0], j = 1, 2, 3. 6.4 Passing to the limit - step 1 Now, combining (6.12), compactness of the embedding (6.13), and the fact that γ > 3/2, we may infer that ̺ε uε ⊗uε → ̺u⊗u weakly in Lq ((0, T )×Ω; R3×3 ) for a certain q > 1. (6.15) Summing up the previous discussion we deduce that the limit functions ̺, u satisfy the equation of continuity Z ̺(τ, ·)ϕ(τ, ·) − ̺0 ϕ(0, ·) dx (6.16) Ω = Z τ 0 Z Ω ̺∂t ϕ + ̺u · ∇x ϕ dx dt for any τ ∈ [0, T ] and any ϕ ∈ Cc∞ ([0, T ] × Ω), together with a relation for the momentum Z ϕ ϕ ̺u(τ, ·) · (τ, ·) − (̺u)0 · (0, ·) dx (6.17) Ω = Z τ 0 Z Ω ̺u · ∂tϕ + ̺u ⊗ u : ∇xϕ + p(̺)divxϕ dx dt 6.5. STRONG CONVERGENCE OF THE DENSITIES − Z τ 0 39 Z ϕ ϕ µ∇x u : ∇x + (λ + µ)divx u divx dx dt Ω for any test function ϕ ∈ Cc∞ ([0, T ] × Ω; R3 ). Here, we have also to assume at least weak convergence of the initial data, specifically, ̺0,ε → ̺0 weakly in Lγ (Ω), (6.18) (̺u)0,ε → (̺u)0 weakly in L1 (Ω; R3 ). Thus it remains to show the crucial relation p(̺) = p(̺) or, equivalently, ̺ε → ̺ a.a. in (0, T ) × Ω. (6.19) This will be carried over in a series of steps specified in the remaining part of this chapter. 6.5 Strong convergence of the densities In order to simplify presentation and to highlight the leading ideas, we assume that γ ≥ 5, in particular ̺ε → ̺ in Cweak (0, T ; Lγ (Ω)), γ ≥ 5. 6.5.1 Renormalized equation We start with the renormalized equation (5.3) with b(̺) = ̺ log(̺) − ̺: Z TZ Z (̺ε log(̺ε ) ∂t ψ − ̺ε divx uε ψ dx dt = − ̺0,ε log(̺0,ε ) dx (6.20) 0 Ω Ω for any ψ ∈ Cc∞ [0, T ), ψ(0) = 1. Clearly, relation (6.20) is a direct consequence of (5.3). Repeating the procedure from Chapter 2 we can get Z Z TZ Z ̺ε log(̺ε )(t, ·) dx + ̺ε divx uε dx dt = ̺0,ε log(̺0,ε ) dx. (6.21) Ω 0 Ω Ω 40 CHAPTER 6. WEAK SEQUENTIAL STABILITY Passing to the limit for ε → 0 in (6.21) and making use of (6.18) we get Z Z tZ Z ̺ log ̺(t, ·) dx + ̺divx u dx dτ = ̺0 log(̺0 ) dx. (6.22) Ω 0 Ω Ω Our next goal is to show that the limit functions ̺, u, besides (6.16), satisfy also its renormalized version. To this end, we use the procedure proposed by DiPerna and Lions [4], specifically, we regularize (6.16) by a family of regularizing kernels κδ (x) to obtain: ∂t ̺δ + divx (̺δ u) = divx (̺δ u) − [divx (̺u)]δ , with vδ = κδ ∗ v, where ∗ stands for spatial convolution. We easily deduce that ∂t b(̺δ ) + divx (b(̺δ )u) + b′ (̺δ )̺δ − b(̺δ ) divx u = b′ (̺δ ) divx (̺δ u) − [divx (̺u)]δ . Taking the limit δ → 0 and using Friedrich’s lemma (see Chapter 7; here we need that ̺ ∈ L2 ((0, T ) × Ω)) and the procedure from Chapter 2 we get Z Z Z tZ ̺divx u dx dτ = ̺0 log(̺0 ) dx; ̺ log(̺)(t, ·) dx + 0 Ω Ω Ω whence, in combination with (6.22), Z tZ Z ̺ log(̺)−̺ log(̺) (t, ·) dx+ ̺divx u−̺divx u dx dτ = 0. (6.23) 0 Ω Ω Assume, for a moment, that we can show Z τZ Z τZ ̺divx u dx dt for any τ > 0, ̺divx u dx dt ≥ 0 Ω 0 (6.24) Ω which, together with lower semi-continuity of convex functionals, yields ̺ log(̺) = ̺ log(̺). In order to continue, we need the following (standard) result: (6.25) 6.5. STRONG CONVERGENCE OF THE DENSITIES Lemma 6.2 Suppose that ̺ε → ̺ weakly in L2 (Ω), where ̺ log(̺) = ̺ log(̺). Then ̺ε → ̺ in L1 (Ω). Proof: Suppose that 0 < δ ≤ ̺. Consequently, because of convexity of z 7→ z log(z), we have a.e. in Ω ̺ε log(̺ε ) − ̺ log(̺) = (log(̺) + 1) (̺ε − ̺) + α(δ)|̺ε − ̺|2 , α(δ) > 0, therefore =− + 1 α(δ) Z 1 α(δ) Z Z {δ≤̺} {δ≤̺} {δ≤̺} |̺ε − ̺|2 dx dt (log(̺) − 1)(̺ε − ̺) dx dt (̺ε log(̺ε ) − ̺ log(̺)) dx dt. Thus we conclude that ̺ε → ̺ a.a. on the set {̺ ≥ δ} for any δ > 0. Now, since and ̺ε → ̺ a.a. on the set {̺ = 0} |{0 < ̺ < δ}| → 0 as δ → 0, we obtain the desired conclusion. 41 42 CHAPTER 6. WEAK SEQUENTIAL STABILITY In accordance with the previous discussion, the proof of strong (pointwise) convergence of {̺ε }ε>0 reduces to showing (6.24). This will be done in the next section. 6.5.2 The effective viscous flux The effective viscous flux (2µ + λ)divx u − p(̺) is a remarkable quantity that enjoys better regularity and compactness properties than its components separately. To see this, we start with the momentum equation Z ̺ε uε (τ, ·) · ϕ (τ, ·) − (̺u)0,ε · ϕ (0, ·) dx (6.26) Ω = Z 0 τ Z ̺ε uε · ∂tϕ + ̺ε (uε ⊗ uε ) : ∇xϕ + p(̺ε )divxϕ dx dt − Z Ω τ 0 Z Ω µ∇x uε : ∇xϕ + (λ + µ)divx uε divxϕ dx dt, together with its weak limit Z ̺u(τ, ·) · ϕ (τ, ·) − (̺u)0 · ϕ (0, ·) dx Ω = Z τ 0 − Z Z ̺u · ∂tϕ + ̺(u ⊗ u) : ∇xϕ + p(̺)divxϕ dx dt Ω τ 0 Z Ω µ∇x u : ∇xϕ + (λ + µ)divx udivxϕ dx dt. Our goal is to take ϕ = ϕε = φ∇x ∆−1 [1Ω ̺ε ], φ ∈ Cc∞ (Ω) as a test function in (6.26), and ϕ = φ∇x ∆−1 [1Ω ̺], φ ∈ Cc∞ (Ω), in (6.27). (6.27) 6.5. STRONG CONVERGENCE OF THE DENSITIES 43 Here, ∆−1 represents the inverse of the Laplacean for space-periodic functions. Since Ω ⊂ R3 is a bounded domain, we have ∇x ∆−1 [1Ω ̺ε ] bounded in L∞ (0, T ; W 1,γ (Ω; R3 )), γ > 3. Moreover, as 1Ω ̺ε as well as 1Ω ̺ satisfy the equation of continuity, we have ∂t ∇x ∆−1 [1Ω ̺ε ] = −∇x ∆−1 divx [̺ε uε ], ∂t ∇x ∆−1 [1Ω ̺] = −∇x ∆−1 divx [̺u]. Step 1: As ̺ε → ̺ in Cweak ([0, T ]; Lγ (Ω)), we have, in accordance with the standard Sobolev embedding relation W 1,γ (Ω) ֒→֒→ C(Ω), ∇x ∆−1 [1Ω ̺ε ] → ∇x ∆−1 [1Ω ̺] in C([0, T ] × Ω). In particular, we deduce from (6.26), (6.27), Z τZ ϕ ϕ ϕ lim ̺ε uε · ∂t ε + ̺ε uε ⊗ uε : ∇x ε + p(̺ε )divx ε dx dt ε→0 0 Ω Z − = τ 0 Z − with Ω τ 0 Z Z µ∇x uε : ∇xϕ ε + (λ + µ)divx uε divxϕ ε dx dt Z ̺u · ∂tϕ + ̺u ⊗ u : ∇xϕ + p(̺)divxϕ dx dt Ω Z µ∇x u : ∇xϕ + (λ + µ)divx udivxϕ dx dt, τ 0 Ω ϕ = φ∇x ∆−1 [1Ω ̺], similarly for ϕ ε . Therefore Z τZ −1 lim φp(̺ε )̺ε + p(̺ε )∇x φ · ∇x ∆ [1Ω ̺ε ] dx dt ε→0 − lim ε→0 0 Z Ω τ 0 Z Ω φ µ∇x uε : ∇2x ∆−1 [1Ω ̺ε ] + (λ + µ)divx uε ̺ε dx dt (6.28) 44 CHAPTER 6. WEAK SEQUENTIAL STABILITY − lim ε→0 Z τ 0 Z µ∇x uε · ∇φ · ∇x ∆−1 [1Ω ̺ε ] Ω −1 +(λ + µ)divx uε ∇x φ · ∇x ∆ [1Ω ̺ε ] dx dt Z τZ φp(̺)̺ + p(̺)∇x φ · ∇x ∆−1 [1Ω ̺] dx dt = − Z − τ 0 Z 0 τ 0 Z Ω Ω φ µ∇x u : ∇2x ∆−1 [1Ω ̺] + (λ + µ)divx u̺ dx dt Z −1 −1 µ∇x u · ∇φ · ∇x ∆ [1Ω ̺] + (λ + µ)divx u∇x φ · ∇x ∆ [1Ω ̺] dx dt Ω + lim ε→0 − Z τ 0 Z Ω Z τ 0 Z φ̺ε uε · ∇x ∆−1 [divx (̺ε uε )] Ω −̺ε uε ⊗ uε : ∇x φ∇x ∆−1 [1Ω ̺ε ] dx dt φ̺u · ∇x ∆−1 [divx (̺u)] − ̺(u ⊗ u) : ∇x φ∇x ∆−1 [1Ω ̺] dx dt. Step 2: We have Z φ∇x uε : Ω ∇2x ∆−1 [1Ω ̺ε ] = Z X 3 Ω i,j=1 − = Z φdivx uε ̺ε dx+ Ω Z X 3 Ω i,j=1 Z Ω dx = Z φ Ω 3 X i,j=1 ∂xj uiε [∂xi ∆−1 ∂xj ][1Ω ̺ε ] dx ∂xj (φuiε )[∂xi ∆−1 ∂xj ][1Ω ̺ε ] dx ∂xj φuiε [∂xi ∆−1 ∂xj ][1Ω ̺ε ] dx ∇x φ·uε ̺ε dx− Z X 3 Ω i,j=1 ∂xj φuiε [∂xi ∆−1 ∂xj ][1Ω ̺ε ] dx. Consequently, going back to (6.28) and dropping the compact terms, we obtain Z τZ lim φ p(̺ε )̺ε − (λ + 2µ)divx uε ̺ε dx dt (6.29) ε→0 0 Ω 45 6.5. STRONG CONVERGENCE OF THE DENSITIES Z − Z τ 0 Z φ p(̺)̺ − (λ + 2µ)divx u̺ dx dt − 0 Ω Z τZ = lim φ ̺ε uε · ∇x ∆−1 [divx (̺ε uε )] ε→0 0 Ω −1 −̺ε (uε ⊗ uε ) : ∇x ∆ ∇x [1Ω ̺ε ] dx dt τ Z φ̺u · ∇x ∆−1 [divx (̺u)] − ̺(u ⊗ u) : ∇x ∆−1 ∇x [1Ω ̺] dx dt. Ω Step 3: Our ultimate goal is to show that the right-hand side of (6.29) vanishes. To this end, we write ̺ε uε · ∇x ∆−1 [divx (̺ε uε )] − ̺ε (uε ⊗ uε ) : ∇x ∆−1 ∇x [1Ω ̺ε ] = uε · ̺ε ∇x ∆−1 [divx (̺ε uε )] − ̺ε uε · ∇x ∆−1 ∇x [1Ω ̺ε ] . Consider the bilinear form 3 X i j i j [v, w] = v Ri,j [w ] − w Ri,j [v ] , Ri,j = ∂xi ∆−1 ∂xj , i,j=1 where we may write 3 X i,j=1 v i Ri,j [wj ] − wi Ri,j [v j ] 3 X = (v i − Ri,j [v j ])Ri,j [wj ] − (wi − Ri,j [wj ])Ri,j [v j ] i,j=1 = U · V − W · Z, where i U = 3 X j=1 i j i (v − Ri,j [v ]), W = and V i = ∂ xi 3 X j=1 ∆−1 ∂xj wj ! 3 X j=1 (wi − Ri,j [wj ]), divx U = divx W = 0, , Z i = ∂ xi 3 X j=1 ∆−1 ∂xj v j ! , i = 1, 2, 3. 46 CHAPTER 6. WEAK SEQUENTIAL STABILITY Thus a direct application of Div-Curl lemma (Lemma 6.1) yields [vε , wε ] → [v, w] weakly in Ls (R3 ) whenever vε → v weakly in Lp (R3 ; R3 ), wε → w weakly in Lq (R3 ; R3 ), and 1 1 1 + = < 1. p q s Seeing that ̺ε → ̺ in Cweak ([0, T ]; Lγ (Ω)), ̺ε uε → ̺u in Cweak ([0, T ]; L2γ/(γ+1) (Ω; R3 )) we conclude that 1Ω ̺ε (t, ·)∇x ∆−1 [divx (̺ε uε )(t, ·)] − (̺ε uε )(t, ·) · ∇x ∆−1 ∇x [1Ω ̺ε (t, ·)] (6.30) → ̺(t, ·)∇x ∆−1 [divx (̺u)(t, ·)] − (̺u)(t, ·) · ∇x ∆−1 ∇x [1Ω ̺(t, ·)] weakly in Ls (Ω; R3 ) for all t ∈ [0, T ], with s= 2γ 6 > since γ ≥ 5. γ+3 5 Thus we conclude that the convergence in (6.30) takes place in the space Lq (0, T ; W −1,2 (Ω)) for any 1 ≤ q < ∞; whence, going back to (6.29), we conclude lim ε→0 Z = τ 0 Z Z τ 0 Ω Z φ p(̺ε )̺ε − (λ + 2µ)divx uε ̺ε dx dt Ω (6.31) φ p(̺)̺ − (λ + 2µ)divx u̺ dx dt. As a matter of fact, using exactly same method and localizing also in the space variable, we could prove that p(̺)̺ − (λ + 2µ)divx u̺ = p(̺)̺ − (λ + 2µ)divx u̺, (6.32) 6.5. STRONG CONVERGENCE OF THE DENSITIES 47 which is the celebrated relation on “weak continuity” of the effective viscous pressure discovered by Lions [12]. Since p is a non-decreasing function, we have Z τZ φ p(̺ε ) − p(̺) (̺ε − ̺) dx dt ≥ 0; 0 Ω now relation (6.31) yields the desired conclusion (6.24), namely Z τZ divx u̺ − divx u̺ dx dt ≥ 0. 0 Ω Thus we get (6.25); whence ̺ε → ̺ a.a. in (0, T ) × Ω and in Lq ((0, T ) × Ω) for any q < γ + min γ2 , 23 γ − 1 . (6.33) 48 CHAPTER 6. WEAK SEQUENTIAL STABILITY Part III Existence of weak solutions for small γ 49 51 Last part of the lecture series is devoted to the proof of existence of weak solutions to the compressible Navier–Stokes system provided p(̺) ∼ ̺γ with γ > 23 . The proof is technically much more complicated than the previous part, however, there are several places which are quite similar to it. Moreover, in the following chapter we also prove several facts (renormalized solution to the continuity equation, continuity in time, estimates of the density etc.) which we skipped in the previous part due to technical complications we tried to avoid there. We first prove the Friedrichs commutator lemma which plays a central role in the study of renormalized solutions to the continuity equation. Next we consider the continuity in time of the density and the momentum. The last chapter contains the core of the existence proof: the approximative problem, existence of a solution for fixed positive regularizing parameters and finally the limit passages which give us solution to our original problem. Note that the proof is performed for the homogeneous Dirichlet boundary conditions for the velocity. The presentation of this part is mostly based on the material from book [14] by A. Novotný and I. Straškraba. 52 Chapter 7 Mathematical tools 7.1 Continuity equation: renormalized solutions and extension We recall that for a function f ∈ Lp (R; Lq (RN )), 1 ≤ p < ∞, 1 ≤ q < ∞, or f ∈ C(R; Lq (RN ) we can define the mollifiers: • over time Tε (f )(t, x) = we have Z R ωε (t − τ )f (τ, x) dτ ; Tε (f ) ∈ C ∞ (R; Lq (RN )), Tε (f ) → f in Lp (R; Lq (RN )) if f ∈ Lp (R; Lq (RN )), Tε (f ) → f in C(I; Lq (RN )) if f ∈ C(R; Lq (RN )), for I a compact subset of R • over space then Sε (f )(t, x) = Z RN ωε (x − y)f (t, y) dy; Sε (f ) ∈ Lp (R; C ∞ (RN )), Sε (f ) → f in Lp (R; Lq (RN )) if f ∈ Lp (R; Lq (RN )) A central technical result is the Friedrichs commutator lemma. 53 54 CHAPTER 7. MATHEMATICAL TOOLS Lemma 7.1 Let N ≥ 2, 1 ≤ q, β ≤ ∞, (q, β) 6= (1, ∞), 1q + β1 ≤ 1. Let 1 ≤ α ≤ ∞, p1 + α1 ≤ 1. Assume for I ⊂ R a bounded time interval ̺ ∈ Lα (I; Lβloc (RN )), 1,q u ∈ Lp (I; Wloc (RN ; RN ). Then in Ls (I; Lrloc (RN )). Sε (u · ∇x ̺) − u · ∇x Sε (̺) → 0 Here, 1s = α1 + p1 and r ∈ [1, q] if β = ∞, and q ∈ (1, ∞], otherwise, where u · ∇x ̺ := divx (̺u) − ̺divx (u) 1 q + 1 β ≤ 1 r ≤1 (in D′ (RN )). Proof: To simplify, we consider only the case β, q < ∞. Step 1: We have D E Sε (u · ∇x ̺), ϕ Z T Z Z ̺(t, y)u(t, y) · ∇x ωε (x − y) dy ϕ(t, x) dx dt = 0 RN RN Z T Z Z ̺(t, y)div u(t, y)ωε (x − y) dy ϕ(t, x) dx dt, − D E u·∇x Sε (̺), ϕ = Therefore D RN RN 0 Z T 0 Z u(t, x)· RN E Sε (u · ∇x ̺) − u · ∇x Sε (̺), ϕ = with Iε (t, x) = and Z RN Z RN T 0 Z ∇x ωε (x−y)̺(t, y) dy ϕ(t, x) dx dt. RN Iε (t, x) − Jε (t, x) ϕ(t, x) dx dt ̺(t, y) u(t, y) − u(t, x) · ∇x ωε (x − y) dy, Jε (t, x) = We define r0 as Z Z RN ̺(t, y)divx u(t, y)ωε (x − y) dy. 1 1 1 = + r0 β q 55 7.1. CONTINUITY EQUATION and get 0 Jε → ̺divx u strongly in Ls (I; Lrloc (RN )). In Steps 2, 3 and 4 we show that 0 Iε → ̺divx u strongly in Ls (I; Lrloc (RN )) which will finish the proof of this lemma. Step 2: We aim at proving kIε kLr0 (BR ) ≤ Ck̺(t)kLβ (BR+1 ) k∇x u(t)kLq (BR+2 ;RN ×N ) for a.a. t ∈ (0, T ). We have kIε krL0r0 (BR ) r0 Z Z 1 x−y dx dy = · ∇ω ̺(t, y) u(t, y) − u(t, x) N +1 ε ε B |x−y|≤ε r0 Z R Z u(t, x − εz) − u(t, x) dx · ∇ω(z) dz = ̺(t, x − εz) ε BR |z|≤1 r Z 0′ r0 r0′ ≤ |∇ω(z)| dz × |z|≤1 r0 Z Z r0 u(t, x − εz) − u(t, x) × |̺(t, x − εz)| dz dx ε BR |z|≤1 r0 Z Z r0 u(t, ξ + εz) − u(t, ξ) ≤ C(ω) |̺(t, ξ)| dz dξ (for ε < 1) ε BR+1 |z|≤1 q r0 Z Z q r0 u(t, ξ + εz) − u(t, ξ) dz ≤ C(ω)|B1 | β |̺(t, ξ)|r0 dξ ε BR+1 |z|≤1 ! rβ0 Z ≤ C(ω, N, q, β) |̺(t, ξ)|β dξ × BR+1 × Z BR+1 Z ! rq0 u(t, ξ + εz) − u(t, ξ) q dz dξ ε |z|≤1 ≤ Ck̺(t)kLβ (BR+1 ) k∇x u(t)kLq (BR+2 ;RN ×N ) . 56 CHAPTER 7. MATHEMATICAL TOOLS Step 3: Let us show the strong convergence, first for a.a. t ∈ (0, T ), in 0 (RN ). Due to Step 2 it is enough to verify that the strong convergence Lrloc holds for any ̺ ∈ Cc∞ (RN ), t ∈ (0, T ) fixed. Indeed, let ̺n ∈ Cc∞ (RN ), ̺n → ̺(t, ·) in Lβ (BR+1 ). Then k(Iε − ̺divx u)(t, ·)kLr0 (BR ) Z ≤ ̺(t, y) − ̺n (y) u(t, y) − u(t, ·) · ∇x ωε (· − y) dy r RN L 0 (BR ) Z + ̺n (y) u(t, y) − u(t, ·) · ∇x ωε (· − y) dy − (̺n divx u(t, ·)) r RN L 0 (BR ) . + ̺n − ̺(t, ·) divx u(t, ·) r L 0 (BR ) The first term is bounded by Ck̺n − ̺(t, ·)kLβ (BR+1 ) k∇x u(t, ·)kLq (BR+2 ;RN ×N ) → 0 for n → ∞, the third is bounded by Ck̺n − ̺(t, ·)kLβ (BR ) kdivx u(t, ·)kLq (BR ;RN ×N ) → 0 for n → ∞. To conclude, let ̺ be a smooth function. Using the change of variables , as above, z = x−y ε Z u(t, x − εz) − u(t, x) · ∇ω(z) dz. Iε (t, x) = ̺(t, x − εz) ε |z|≤1 1,q As u ∈ Wloc (RN ; RN ) for a.a. t ∈ (0, T ), Z 1 u(t, x − εz) − u(t, x) ∇x u(t, x − ετ z) dτ → −z · ∇x u(t, x) = −z · ε 0 for a.a. t ∈ (0, T ) and a.a. (x, z) ∈ RN × B1 (a.a. points are Lebesgue points). Moreover, as ̺ is smooth, ̺(t, x − εz) → ̺(t, x), (x, z) ∈ BR+1 × B1 , t ∈ (0, T ). Therefore, by Vitali’s theorem Z Z Z (Iε ϕ)(t, x) dx → − ̺(t, x)∂i uj (t, x)ϕ(t, x) dx zi ∂j ω(z) dz B1 RN RN Z (̺divx u)(t, x)ϕ(t, x) dx. = RN 57 7.1. CONTINUITY EQUATION Step 4: We have kIε ksLs (I;Lr0 (BR )) ≤ C Z T 0 k̺(t, ·)ksLβ (BR+1 ) k∇x u(t, ·)ksLq (BR+2 ;RN ×N ) dt ≤ Ck̺ksLα (I;Lβ (BR+1 )) k∇x uksLp (I;Lq (BR+2 ;RN ×N )) . Due to this and the fact that 0 (RN ) for a.a. t ∈ (0, T ), Iε → ̺divx u in Lrloc we get due to Step 2 by the Lebesgue dominated convergence theorem the claim of the lemma. Next we show that we may extend (sufficiently regular) solution to the continuity equation outside a Lipschitz domain in such a way that the extension (by zero in the case of our boundary conditions) solves the continuity equation in the full RN . In particular, this shows that we may use as test functions smooth functions up to the boundary. Lemma 7.2 Let Ω be a bounded Lipschitz domain in RN , N ≥ 2. Let ̺ ∈ L2 (QT ), u ∈ L2 (I; W01,2 (Ω; RN )) and f ∈ L1 (QT ) satisfy ∂t ̺ + divx (̺u) = f in D′ (QT ). Extending (̺, u, f ) by (0, 0, 0) outside Ω, ∂t ̺ + divx (̺u) = f in D′ (I × RN ). Proof: We have to show (after the extension by (0, 0, 0)) Z TZ Z TZ ̺u · ∇x η dx dt ̺∂t η dx dt − − 0 RN 0 RN Z TZ f η dx dt ∀η ∈ Cc∞ ((0, T ) × RN ). = 0 RN Denote Φm ∈ Cc∞ (Ω), m ∈ N, 0 ≤ Φm ≤ 1, 1 Φm (x) = 1 for x ∈ y ∈ Ω; dist(y, ∂Ω) ≥ , m |∇x Φm (x)| ≤ 2m for x ∈ Ω. 58 CHAPTER 7. MATHEMATICAL TOOLS Evidently, Φm → 1 pointwise in Ω and for any fixed compact K ⊂ Ω, supp ∇x Φm ⊂ Ω \ K for m ≥ m0 (K) ∈ N, We can write Z Z TZ f η dx dt = RN 0 Z T 0 Z RN Z T 0 ̺∂t η dx dt = Z Z 0 T 0 Z T Z RN RN f ηΦm dx dt + Z | ̺∂t (Φm η) dx dt + Z T Z |supp ∇x Φm | → 0. Z T 0 RN f η(1 − Φm ) dx dt, {z } →0 Z | T 0 Z RN ̺∂t η(1 − Φm ) dx dt, {z } →0 ̺u · ∇x (Φm η) dx dt ̺u · ∇x η dx dt = 0 RN Z TZ Z TZ ̺u · ∇x Φm η dx dt. ̺u · ∇x η(1 − Φm ) dx dt − + 0 RN 0 RN | {z } RN →0 We know that Z Z TZ f ηΦm dx dt = − 0 RN T 0 Z RN ̺∂t (Φm η) dx dt− Z T 0 Z RN ̺u·∇x (Φm η) dx dt as Φm η has support in Ω. Therefore we have to show that Z TZ ̺u · ∇x Φm η dx dt → 0. Im = RN 0 But due to the Hardy inequality Z TZ |̺||u||∇x Φm ||η| dx dt |Im | ≤ 0 RN Z T k̺kL2 ({supp ∇x Φm }) ≤ 2 sup |η(t, x)| t,x ≤ C(η, Ω) Z 0 T 0 u dt dist(x, ∂Ω) L2 (Ω;RN ) k̺kL2 ({supp ∇x Φm }) kukW 1,2 (Ω;RN ) dt 0 ≤ C(η, Ω)k̺kL2 (0,T ;L2 ({supp ∇x Φm })) kukL2 (0,T ;W 1,2 (Ω;RN )) → 0 0 as m → ∞. The lemma is proved. 59 7.1. CONTINUITY EQUATION Remark 7.1 Hence, in case ̺ ∈ L2 ((0, T )×Ω) (as u ∈ L2 (0, T ; W01,2 (Ω; R3 )) will be satisfied), we further have Z Z ̺(t, x) dx = ̺(s, x) dx for any t, s ∈ [0, T ]. Ω Ω It is enough to take η ≡ 1 in [s, t] × Ω, provided ̺ is weakly continuous in L1 (Ω) (which will be proved later). On the other hand, if only ̺ ∈ Lp (QT ), 1 ≤ p < 2, the mass may not be conserved. An explicit counterexample (due to E. Feireisl and H. Petzeltová) shows this, even in 1D. Let Ω = (0, 1) and Z x α 1 1 1 h t− dy , < α < 1, u(x) = x(1 − x) , ̺(t, x) = u(x) 2 0 u(y) with h ∈ C 1 (R), h(s) = 0 for s ≤ 0. Evidently ( ′ u ∼ xα−1 , x → 0, =⇒ α > 21 , 1,2 u ∈ W0 (0, 1), u′ ∼ (1 − x)α−1 , x → 1, =⇒ α > 12 , 1 1 1 p p ̺ ∈ C([0, T ]; L (0, 1)), 1 ≤ p < ∈L ,p< ⇒ α < 1, α u α and Rx 1 t − 0 u(y) dy ⇒ ∂t ̺ + ∂x (̺u) = 0. Rx 1 −1 dy u(x) ∂x (̺u) = h′ t − 0 u(y) ∂t ̺ = But 1 h′ u(x) Z ̺(t, x) dx is not constant, as Z Z ̺(0, x) dx = 0, but for h suitably chosen ̺(t, x) dx 6= 0 ∀t > 0. Ω Ω Ω This example can also be generalized to higher space dimensions. We finish this section by showing that, under certain regularity assumptions, a weak solution to the continuity equation is also a renormalized solution. This fact will be important in the proof of the existence of weak solution in the last chapter. Due to Lemma 7.1 we have 60 CHAPTER 7. MATHEMATICAL TOOLS Lemma 7.3 Let N ≥ 2, 2 ≤ β < ∞, λ0 < 1, −1 < λ1 ≤ b ∈ C([0, ∞)) ∩ C 1 ((0, ∞)), β 2 − 1 and |b′ (t)| ≤ ct−λ0 , t ∈ [0, 1], |b′ (t)| ≤ ctλ1 , t ≥ 1. (7.1) (7.2) 1,2 Let ̺ ∈ Lβ (I; Lβloc (RN )), ̺ ≥ 0 a.e. in I × RN , u ∈ L2 (I; Wloc (RN ; RN )) and β if λ1 > 0, z = 1 if λ1 ≤ 0. Suppose that f ∈ Lz (I; Lzloc (RN )), z = β−λ 1 ∂t ̺ + divx (̺u) = f in D′ (I × RN ). (7.3) (i) Then for any b ∈ C 1 ([0, ∞)) satisfying (7.2) we have ∂t b(̺) + divx (b(̺)u) + {̺b′ (̺) − b(̺)}divx u = f b′ (̺) in D′ (I × RN ). (7.4) (ii) If f = 0, then (7.4) holds for any b satisfying (7.1) and (7.2). Proof: We consider only the case (ii), leaving (i) as possible exercise for the reader. We regularize (7.3) over space variable and get ∂t Sε (̺) + divx (Sε (̺)u) = rε (̺, u) a.e. in I × RN , (7.5) where rε (̺, u) = divx (Sε (̺)u) − divx (Sε (̺u)). But rε (̺, u) = u · ∇x Sε (̺) + Sε (̺)divx u − Sε (divx (̺u)) = u · ∇x Sε (̺) − Sε (u · ∇x ̺) + Sε (̺)divx u − Sε (̺divx u), hence by Lemma 7.1 and an easy observation rε (̺, u) → 0 in Lr (I; Lrloc (RN )), 1 1 1 = + (≤ 1). r β 2 To avoid singularity at ̺ = 0, we multiply (7.5) by b′h (Sε (̺)) with bh (·) = b(h + ·), h > 0, and obtain ∂t bh (Sε (̺)) + divx (bh (Sε (̺))u) + Sε (̺)b′h (Sε (̺)) − bh (Sε (̺)) divx u = rε b′h (Sε (̺)) a.e. in I × RN . 61 7.1. CONTINUITY EQUATION Now we pass with ε → 0+ . As Sε (̺) → ̺ in Lβ (I; Lβloc (RN )) (i.e. for a subsequence a.e. in I × RN ), we get by Vitali’s (convergence) theorem that bh (Sε (̺)) → bh (̺), Sε (̺)b′h (Sε (̺)) − bh (Sε (̺)) → ̺b′h (̺) − bh (̺) in Lploc (I × RN ), 1 ≤ p < 2 β (Sε (̺)b′h (Sε (̺)) ≤ CSε (̺)1+ 2 −1 for Sε (̺) ≫ 1). As this term is bounded also in L2 (I × Ω′ ) for Ω′ bounded subset of RN , then the convergence holds also in L2weak (I × Ω′ ). Therefore, passing with ε → 0, we have ∂t bh (̺) + divx (bh (̺)u) + ̺b′h (̺) − bh (̺) divx u = 0, as Z T 0 Z Ω′ rε b′h (Sε (̺)) dx dt ≤ Z T 0 krε kLr (Ω′ ) kb′h (Sε (̺))kLr′ (Ω′ ) dt → 0, and kb′h (Sε (̺))kLr (Ω′ ) ≤ kSε (̺)kLβ (Ω′ ) . where r1′ = 1 − 1r = 21 − β1 = β−2 2β Finally we aim to pass with h → 0+ . Recall that |{(t, x); ̺ ≥ k} ∩ (I × Ω′ )| ≤ k −β k̺kβLβ ((I×Ω′ )∩{̺≥k}) . Then we write for ψ ∈ Cc∞ (I × RN ) Z ̺b′h (̺) − bh (̺) divx uψ dx dt I×RN Z = ̺b′h (̺) − bh (̺) divx uψ dx dt (I×RN )∩{̺≤k}∩supp ψ Z + ̺b′h (̺) − bh (̺) divx uψ dx dt. (I×RN )∩{̺>k}∩supp ψ Now, passing with h → 0+ , the first term on the right-hand side goes to Z I×RN ′ ̺b (̺) − b(̺) divx uψ1{̺≤k} dx dt, 62 CHAPTER 7. MATHEMATICAL TOOLS due to the Lebesgue dominated convergence theorem. The second term can be controlled by Z β β −1 2 2 + ̺ |divx u| |ψ| dx dt C ̺(̺ + h) {̺>k} Z β ≤C ̺ 2 + ̺ |divx u||ψ| dx dt {̺≥k} β ≤ C k̺kL2 β ((I×Ω′ )∩{̺≥k}) kdivx ukL2 (I×Ω′ ) β 2 + k̺kLβ ((I×Ω′ )∩{̺≥k}) k Further, Z ̺b′ (̺) − b(̺) divx uψ1{̺≤k} dx dt I×RN Z →k→∞ 1− β2 I×RN kdivx uk L2 (I×Ω′ ) →k→∞ 0. ̺b′ (̺) − b(̺) divx uψ dx dt, by the same argument. The other terms can be controlled similarly. The lemma is proved. 7.2 Continuity in time First, we have Definition 7.1 The function g belongs to Cweak ([0, T ]; Lq (Ω)), 1 ≤ q < ∞, R ′ if Ω gϕ dx ∈ C([0, T ]) for all ϕ ∈ Lq (Ω). We have the following very easy result: Lemma 7.4 Let 1 < q < ∞, Ω ⊂ RN , I ⊂ R be bounded. Let f ∈ R L∞ (I; Lq (Ω)) and ∂t Ω f η dx ∈ L1 (I) for all η ∈ Cc∞ (Ω). Then there exists g ∈ Cweak (I; Lq (Ω)) such that for a.a. t ∈ I f (t, ·) = g(t, ·) (in the sense of Lq (Ω)). R Proof: Take any η ∈ Cc∞ (Ω). As Ω f η dx ∈R W 1,1 (I), (after a possible change on times of measure zero) we have that Ω f η dx ∈ C(I). Then it is easy to see that sup kf (t, ·)kLq (Ω) ≤ ess sup kf (t, ·)kLq (Ω) . t∈I t∈I 63 7.2. CONTINUITY IN TIME ′ ′ Choose ε > 0 and take arbitrary ϕ ∈ Lq (Ω). Since Cc∞ (Ω) is dense in Lq (Ω), 1 ≤ q ′ < ∞, we have Z Z f (t + δ, ·) − f (t, ·) ϕ dx ≤ f (t + δ, ·) − f (t, ·) η dx Ω Ω Z + f (t + δ, ·) − f (t, ·) (ϕ − η) dx , Ω where η ∈ Cc∞ (Ω) is suitably chosen in such a way R that the second integral is less than ε/2. Now, due to the continuity of Ω f η dx, we can choose δ0 sufficiently small that for 0 ≤ |δ| ≤ δ0 the first integral is bounded by ε/2. The lemma is proved. . Remark 7.2 Looking at the weak formulation of the continuity equation, as ̺ ∈ L∞ (0, T ; Lγ (Ω)) and u ∈ L2 (0, T ; W01,2 (Ω; R3 )), we immediately see (at least for γ > 56 ) that ̺ ∈ Cweak ([0, T ]; Lγ (Ω)), as Z Z ∂t ̺η dx = − ̺u · ∇x η dx ∈ L1 (I), I ⊂ [0, T ). Ω Ω 2γ Remark 7.3 Similarly we have that ̺u ∈ Cweak ([0, T ]; L γ+1 (Ω; R3 )). Indeed, Z (̺|u|) Ω 2γ γ+1 dx = Z γ γ (̺|u|2 ) γ+1 ̺ γ+1 dx Ω ≤ Z 2 ̺|u| dx Ω γ Z γ+1 γ ̺ dx Ω 1 γ+1 ∈ L∞ (I). Looking at the weak formulation of the momentum equation, it is an easy task to verify that for ϕ ∈ Cc∞ ([0, T ] × Ω; R3 ) Z ∂t ̺u · ϕ dx ∈ L1 (I), Ω which finishes the proof. In what follows we will use the following abstract version of the Arzelà– Ascoli theorem (see [10, Theorem 1.6.9]) 64 CHAPTER 7. MATHEMATICAL TOOLS Theorem 7.1 Let X and B be Banach spaces such that B ֒→֒→ X. Let fn be a sequence of functions: I → B which is uniformly bounded in B and uniformly continuous in X. Then there exists f ∈ C(I; B) such that fn → f in C(I; X) at least for a chosen subsequence. Then we have Theorem 7.2 Let 1 < p, q < ∞, Ω be a bounded Lipschitz domain in RN , q N ≥ 2. Let {gn }∞ n=1 be a sequence of functions: I → L (Ω) such that • gn ∈ Cweak (I; Lq (Ω)) • gn is uniformly continuous in W −1,p (Ω) • gn is uniformly bounded in Lq (Ω). Then at least for a chosen subsequence (i) gn → g in Cweak (I; Lq (Ω)). (ii) If moreover Lq (Ω) ֒→֒→ W −1,p (Ω) (i.e. 1 < p ≤ p N < p < ∞, NN+p < q < ∞), then N −1 N N −1 and 1 < q < ∞, or in C(I; W −1,p (Ω)). n o Proof: (i) As W −1,p (Ω) ֒→ W −1,s (Ω) for s = min p, NN−1 , the sequence gn gn → g is uniformly continuous in W −1,s (Ω). As the embedding Lq (Ω) ֒→ W −1,s (Ω) is compact, we have by virtue of Theorem 7.1 gn → g in C(I; W −1,s (Ω)), at least for a chosen subsequence. Therefore, for a given ε > 0 there exists n0 such that for m, n > n0 : Z (gn (t, ·) − gm (t, ·))η dx Ω ≤ k(gn (t, ·) − gm (t, ·)kW −1,s (Ω) kηkW 1,s′ (Ω) ≤ εkηkW 1,s′ (Ω) , ∞ ∞ for R all η ∈ Cc (Ω), for all t ∈ I. Hence for any η ∈ Cc (Ω) the mappings t 7→ g (t, ·)η dx form a Cauchy sequence in C(I) which has a limit Aη ∈ C(I). Ω n As Z sup |Aη (t)| ≤ lim sup gn (t, ·)η dx ≤ CkηkLq′ (Ω) , t∈I n→∞ Ω 65 7.2. CONTINUITY IN TIME η ∈ Cc∞ (Ω), we see that η 7→ Aη is a linear densely defined bounded operator from Lq (Ω) to R. Hence Z Aη (t) = g(t, ·)η dx with g(t, ·) ∈ Lq (Ω). Ω R Moreover, t 7→ Ω g(t, ·)η dx ∈ C(I) for all η ∈ Cc∞ (Ω) and by the density ′ argument also for η ∈ Lq (Ω). Moreover, again by the density argument Z sup (gn (t, ·) − g(t, ·))η dx →n→∞ 0, t∈I hence ′ Z Ω Ω gn (t, ·)η dx → Z Ω g(t, ·)η dx in C(I) for any η ∈ Lq (Ω). To prove (ii), recall that Lq (Ω) ֒→֒→ W −1,p (Ω) and the result follows from Theorem 7.1. Next Lemma 7.5 Let Ω be a bounded Lipschitz domain in RN , N ≥ 2, 1 < q < ∞, 1 ≤ p < ∞. If gn → g in Cweak (I; Lq (Ω)), then gn → g strongly in Lp (I; W −1,r (Ω)) provided Lq (Ω) ֒→֒→ W −1,r (Ω). Proof: As gn (t, ·) ⇀ g(t, ·) and Lq (Ω) ֒→֒→ W −1,r (Ω), we have gn (t, ·) → g(t, ·) in Lq (Ω), t ∈ I, in W −1,r (Ω), t ∈ I. As in particular gn is bounded in L∞ (I; Lq (Ω)), then also sup kgn (t, ·)kLq (Ω) ≤ C t∈I and so is supt∈I kgn (t, ·)kW −1,r (Ω) . Thus by the Lebesgue dominated convergence theorem Z T k(gn (t, ·) − g(t, ·)kpW −1,r (Ω) dt →n→∞ 0. 0 We further have 66 CHAPTER 7. MATHEMATICAL TOOLS Lemma 7.6 Let N ≥ 2, 1 < β < ∞, θ ∈ (0, β4 ) and Ω be a bounded domain in RN . Let the pair (̺, u) fulfill ̺ ≥ 0 a.e. in (0, T ) × RN , ̺ ∈ L∞ (0, T ; Lβloc (RN )) ∩ Cweak ([0, T ]; Lβ (Ω)), 1,2 u ∈ L2 (0, T ; Wloc (RN ; RN )) and let (̺, u) solve the renormalized continuity equation with b(s) = sθ , i.e. ∂t ̺θ + divx (̺θ u) + (θ − 1)̺θ divx u = 0 in D′ ((0, T ) × RN ). (7.6) Then ̺ ∈ C([0, T ]; Lp (Ω)), 1 ≤ p < β. Remark 7.4 In our case we have ̺ ∈ C([0, T ]; Lp (Ω)), 1 ≤ p < γ. R Proof: Due to (7.6) we know that ∂t Ω ρθ η dx ∈ L2 (0, T ), hence by Lemma β 7.4 we know that ̺ = ̺e a.e., where ̺eθ ∈ Cweak ([0, T ]; L θ (Ω)). We now take (7.6) with ̺e and regularize it over the space variable by the mollifier Sε . Thus ∂t Sε (e ̺θ )+divx (Sε (e ̺θ )u) = (1−θ)Sε (e ̺θ divx u)+rε (e ̺θ , u) in D′ ((0, T )×RN ), (7.7) where rε (e ̺θ , u) = divx (Sε (e ̺θ )u) − divx (Sε (e ̺θ u)). Indeed, Sε (e ̺θ ) ∈ C([0, T ] × Ω), kSε (e ̺θ )(t, ·)kLq (RN ) ≤ ke ̺θ (t, ·)kLq (RN ) by the Hausdorff-Young inequality. Therefore there exists ε0 > 0 such that sup sup kSε (e ̺θ )(t, ·)kLq (RN ) < ∞, ε∈(0,ε0 ) t∈[0,T ] 1≤q≤ β . θ Furthermore, Sε (e ̺θ )(t, ·) → ̺eθ (t, ·) strongly in Lq (Ω), 1 ≤ q ≤ βθ , t ∈ [0, T ], 2β Sε (e ̺θ divx u) → ̺eθ divx u strongly in L2 (0, T ; L 2θ+β (Ω)). (7.8) By Lemma 7.1 (Friedrichs commutator lemma) 2β rε (e ̺θ , u) → 0 in L2 (I; L 2θ+β (Ω)). We now apply Lemma 7.3 (renormalized solution with non-zero right hand side) with b(s) = (s + 1)2 to (7.7) ∂t (Sε (e ̺θ ) + 1)2 + divx ((Sε (e ̺θ ) + 1)2 u) + (Sε (e ̺θ )2 − 1)divx u = 2(1 − θ)(Sε (e ̺θ ) + 1)Sε (e ̺θ divx u) + 2(Sε (e ̺θ ) + 1)rε (e ̺θ , u) (7.9) 67 7.2. CONTINUITY IN TIME R ̺θ )|2 η dx}ε>0 is uniformly bounded in D′ ((0, T ) × RN ). We have that { Ω |Sε (e for every η ∈ Cc∞ (Ω) on [0, T ] and by (7.9) together with assumptions on ̺, u also uniformly continuous on [0, T ]. Now, due to (7.8) and Arzelà-Ascoli theorem Z Z θ 2 |Sε (e ̺ )| η dx → |e ̺θ |2 η dx in C[0, T ], η ∈ Cc∞ (Ω). Ω Ω Therefore, by density argument (ηε → 1), ̺eθ ∈ C([0, T ]; L2 (Ω)). Now, due to interpolation of Lebesgue spaces ̺e ∈ C([0, T ]; Lp (Ω)), 1 ≤ p < β. Finally, due to our assumption ̺ = ̺e for every t ∈ [0, T ]. 68 CHAPTER 7. MATHEMATICAL TOOLS Chapter 8 Existence proof 8.1 Approximations Recall that we aim at proving the existence of weak solutions (in the sense as presented in Chapter 5) to the following problem (we set a = 1 for the sake of simplicity): ∂t (̺u) + divx (̺u ⊗ u) − µ∆u − (µ + λ)∇x divx u +∇x ̺γ = ̺f in (0, T ) × Ω, ∂t ̺ + divx (̺u) = 0 in (0, T ) × Ω, u(t, x) = 0 at (0, T ) × ∂Ω, ̺(0, x) = ̺0 (x), (̺u)(0, x) = (̺u)0 (x) in Ω. (8.1) At the first level we regularize the pressure (δ > 0) and get the regularized system with artificial pressure ∂t (̺u) + divx (̺u ⊗ u) − µ∆u − (µ + λ)∇x divx u +∇x ̺γ + δ∇x ̺β = ̺f in (0, T ) × Ω, ∂t ̺ + divx (̺u) = 0 in (0, T ) × Ω, u(t, x) = 0 at (0, T ) × ∂Ω, ̺(0, x) = ̺0,δ (x), (̺u)(0, x) = (̺u)0,δ (x) in Ω. (8.2) At the next level we regularize the continuity equation (ε > 0) and get 69 70 CHAPTER 8. EXISTENCE PROOF the continuity equation with dissipation ∂t (̺u) + divx (̺u ⊗ u) − µ∆u − (µ + λ)∇x divx u +∇x ̺γ + δ∇x ̺β + ε(∇x ̺ · ∇x )u = ̺f in (0, T ) × Ω, ∂t ̺ + divx (̺u) − ε∆̺ = 0 in (0, T ) × Ω, ∂̺ u(t, x) = 0, = 0 at (0, T ) × ∂Ω, ∂n ̺(0, x) = ̺0,δ (x), (̺u)(0, x) = (̺u)0,δ (x) in Ω. (8.3) The last level is based on the finite dimensional projection (Galerkin approximation) of the momentum equation. We take a basis in W01,2 (Ω; R3 ) (orthogonal) which is orthonormal in L2 (Ω; R3 ) and is formed by eigenfunctions of the Lamé equation Φj − (µ + λ)∇x divxΦ j = αj Φ j , −µ∆Φ 0 < α1 < α2 ≤ . . . , Φ j ∈ W01,p (Ω; R3 ) ∩ W 2,p (Ω; R3 ), 1 ≤ p < ∞ arbitrary, with the scalar products Z µ∇x u : ∇x v + (µ + λ)divx u divx v dx, (u, v)W 1,2 (Ω;R3 ) := 0 Ω Z (u, v)L2 (Ω;R3 ) := u · v dx. Ω We first show existence of solutions to the Galerkin approximation (Section 8.2). Then we collect estimates independent of the dimension of the Galerkin approximation and pass in Section 8.3 with n → ∞. We receive system (8.3), i.e. system with continuity equation with dissipation. Next we prove estimates independent of the parameter ε and pass with ε → 0+ and get the system with the artificial pressure (8.2) (Section 8.4). In the last section we collect estimates independent of δ and pass with δ → 0+ to get a solution to the original system (8.1). 8.2 Existence for the Galerkin approximation We take δ, ε > 0, n ∈ N and β > 1 sufficiently large (e.g. β ≥ 15 is enough). Φ1 , . . . , Φ n }. Our aim is to show: Let us denote Xn = Lin{Φ Theorem 8.1 Under the assumption of Theorem 8.3, let 0 < ̺ ≤ ̺0,δ ≤ ̺ < ∞, ̺0,δ ∈ C ∞ (Ω). Then there exists a (unique) couple (̺n , un ) such that: 8.2. EXISTENCE FOR THE GALERKIN APPROXIMATION 71 (i) ̺n ∈ C([0, T ]; W 1,p (Ω)) ∩ Lp (I; W 2,p (Ω)), ∂t ̺n ∈ Lp (I; Lp (Ω)), 1 ≤ p < ∞, ̺ > 0 a.e. in (0, T ) × Ω, un ∈ C 0,1 ([0, T ]; Xn ) (ii) Z T 0 Z Ω ∂t (̺n un ) · Φ − ̺n (un ⊗un ) : ∇x Φ + µ∇x un : ∇x Φ + (µ + λ)divx un divx Φ − (̺γn + δ̺βn )divx Φ Z TZ ̺n f · Φ dx dt ∀Φ ∈ Xn + ε∇x ̺n ∇x un · Φ dx dt = 0 Ω (iii) ∂t ̺n + divx (̺n un ) − ε∆̺n = 0 a.e. in (0, T ) × Ω n (iv) ̺n (0) = ̺0,δ , un (0) = Pn u0 , ∂̺ | = 0, where Pn is the projector of ∂n ∂Ω 3 2 ∞ L (Ω; R ) to Xn and ̺0,δ ∈ C (Ω) is the regularized initial condition (v) denoting Eδ (̺, u)(t) = Z Ω ̺γ δ̺β 1 ̺|u|2 + + 2 γ−1 β−1 (t, ·) dx, we have Z tZ Eδ (̺n , un )(t) + µ|∇x un |2 + (µ + λ)(divx un )2 dx dτ 0 Ω Z tZ Z tZ 2 γ−2 2 ̺β−2 ̺n |∇̺n | dx dτ + εδβ +εγ n |∇x ̺n | dx dτ 0 Ω Z0t ZΩ ̺n f · un dx dτ + Eδ (̺0,δ , Pn u0 ) a.e. in (0, T ) ≤ 0 (8.4) Ω Let us first look at the parabolic Neumann problem ∂t ̺ − ε∆̺ = h in (0, T ) × Ω, ̺(0) = ̺0 in Ω, ∂̺ = 0 in (0, T ) ∂n ∂Ω (8.5) with h and ̺0 given sufficiently regular functions. We have the following result (for the proof see e.g. [1]) 72 CHAPTER 8. EXISTENCE PROOF Lemma 8.1 Let 0 < θ ≤ 1, 1 < p, q < ∞, Ω bounded, Ω ∈ C 2,θ , ̺0 ∈ k·k 2 f 2− p2 ,q (Ω) = {z ∈ C ∞ (Ω); ∂z |∂Ω = 0} W 2− p ,q (Ω) , where k · k 2− 2 ,q W is the ∂n p W (Ω) norm in the Sobolev-Slobodetskii space. Let h ∈ Lp (0, T ; Lq (Ω)). Then 2 there exists unique ̺ ∈ Lp (0, T ; W 2,q (Ω))∩C([0, T ]; W 2− p ,q (Ω)) with the time derivative ∂t ̺ ∈ Lp (0, T ; Lq (Ω)), together with the estimates 1 ε1− p k̺k + k∂t ̺kLp (0,T ;Lq (Ω)) + εk̺kLp (0,T ;W 2,q (Ω)) 1− p1 ≤ C(p, q, Ω) ε k̺0 k 2− p2 ,q + khkLp (0,T ;Lq (Ω)) . L∞ (0,T ;W 2 ,q 2− p (Ω)) W (Ω) If h = divx b, b ∈ Lp (0, T ; Lq (Ω; R3 )), ̺0 ∈ Lq (Ω), then there exists unique ̺ ∈ Lp (0, T ; W 1,q (Ω)) ∩ C([0, T ]; Lq (Ω)), solving Z Z Z d ̺η dx+ε ∇x ̺·∇x η dx = − b·∇x η dx, η ∈ C ∞ (Ω) in D′ (0, T ) dt Ω Ω Ω and 1 ε1− p k̺kL∞ (0,T ;Lq (Ω)) + εk∇x ̺kLp (0,T ;Lq (Ω)) 1 ≤ C(p, q, Ω) ε1− p k̺0 kLq (Ω) + kbkLp (0,T ;Lq (Ω;R3 )) . We now return to ∂t ̺ + divx (̺u) − ε∆̺ = 0 in (0, T ) × Ω, ̺(0) = ̺0,δ in Ω, ∂̺ = 0 on (0, T ) ∂n ∂Ω (8.6) We aim at proving Lemma 8.2 Let 0 < θ ≤ 1, Ω ∈ C 2,θ bounded, 0 < ̺ ≤ ̺0,δ ≤ ̺ < ∞, ̺0,δ ∈ C ∞ (Ω). Let u ∈ L∞ (0, T ; W01,∞ (Ω; R3 )), where W01,∞ (Ω; R3 ) = {z ∈ W 1,∞ (Ω; R3 ); z|∂Ω = 0}. Then there exists unique solution to (8.6) ̺ = ̺(u) ∈ Lp (0, T ; W 2,p (Ω)) ∩ C([0, T ]; W 1,p (Ω)), ∂t ̺ ∈ Lp (0, T ; Lp (Ω)), 1 < p < ∞, arbitrary. Moreover ̺e− Rt 0 ku(τ )kW 1,∞ (Ω;R3 ) dτ ≤ ̺(t, x) ≤ ̺e Rt 0 ku(τ )kW 1,∞ (Ω;R3 ) dτ , (8.7) 8.2. EXISTENCE FOR THE GALERKIN APPROXIMATION 73 for t ∈ [0, T ] and a.a. x ∈ Ω. If kukL∞ (I;W 1,∞ (Ω;R3 )) ≤ K, then C 2 k̺kL∞ (0,t;W 1,2 (Ω)) ≤ Ck̺0,δ kW 1,2 (Ω) e ε (K+K )t , C C√ 2 k∇2x ̺kL2 ((0,t)×Ω;R3×3 ) ≤ tk̺0,δ kW 1,2 (Ω) Ke ε (K+K )t + k̺0,δ kW 1,2 (Ω) , √ε C 2 k∂t ̺kL2 ((0,t)×Ω) ≤ C tk̺0,δ kW 1,2 (Ω) Ke ε (K+K )t + k̺0,δ kW 1,2 (Ω) , k(̺(u1 ) − ̺(u2 ))kL2 ((0,t)×Ω) ≤ C(K, ε, T )tk̺0,δ kW 1,2 (Ω) ku1 − u2 kL∞ (0,t;W 1,∞ (Ω;R3 )) , (8.8) where t ∈ [0, T ]. Proof: Step 1: First, if Ω ∈ C 2 , u ∈ L∞ (0, T ; W01,∞ (Ω; R3 )), ̺0,δ ∈ W 1,2 (Ω), there exists unique ̺ ∈ C([0, T ]; W 1,2 (Ω)) ∩ L2 (0, T ; W 2,2 (Ω)), ∂t ̺ ∈ L2 ((0, T ) × Ω) solution to (8.6). • We construct the solution by the Galerkin method, with the orthonormal (in L2 ) and orthogonal (in W 1,2 ) basis of the Laplace equation with the Neumann boundary condition at ∂Ω. • For n ∈ N, testing by ̺n , ∆̺n , ∂t ̺n we get k̺n (t)kL∞ (0,T ;W 1,2 (Ω)) ≤ C(T, kukL∞ (I;W 1,∞ (Ω;R3 )) , ε), k∇x ̺n kL2 (I;W 1,2 (Ω;R3 )) ≤ C(T, kukL∞ (I;W 1,∞ (Ω;R3 )) , ε), k∂t ̺n kL2 ((0,T )×Ω) ≤ C(T, kukL∞ (I;W 1,∞ (Ω;R3 )) , ε). • Passing with n → ∞ in Z T Z =− Z T 0 leads to Z Z 0 Ω ∂t ̺n ψ dx z dt + ε Ω Z T 0 Z Ω ∇x ̺n · ∇x ψz dx dt divx (̺n u)ψz dx dt ∀ψ ∈ Lin{h1 , . . . , hn }, z ∈ Cc∞ (0, T ) T 0 Z Ω ∂t ̺ψ dx z dt + ε =− Z T 0 Z T 0 Z Ω Z ∇x ̺ · ∇x ψz dx dt divx (̺u)ψz dx dt Ω for any z ∈ Cc∞ (0, T ) and ψ ∈ Lin{h1 , h2 , . . . }. 74 CHAPTER 8. EXISTENCE PROOF • By the density argument =− Z T 0 Z Z T 0 Ω Z ∂t ̺η dx dt + ε Ω Z T 0 Z Ω ∇x ̺ · ∇x η dx dt divx (̺u)η dx dt ∀η ∈ L2 (0, T ; W 1,2 (Ω)). • Finally, the continuity in W 1,2 (Ω) follows by standard arguments. Step 2: Now, let Ω ∈ C 2,θ . We apply Lemma 8.1 (with the right-hand side h := divx (̺u) ∈ L2 (0, T ; L6 (Ω)) ∩ L∞ (0, T ; L2 (Ω)) and get (in two steps) that ̺ ∈ Lp (0, T ; W 2,p (Ω)) ∩ C([0, T ]; W 1,p (Ω)) with ∂t ̺ ∈ Lp ((0, T ) × Ω) for any 1 < p < ∞. Rt Step 3: Consider R(t) = ̺e 0 kdivx u(τ,·)kL∞ (Ω) dτ . Then R′ (t) − kdivx u(t, ·)kL∞ (Ω) R(t) = 0, R(0) = ̺ and R′ + divx (Ru) ≥ 0 a.e. in QT . Denote ω(t, x) = ̺(t, x) − R(t). Then ∂t ω + divx (ωu) − ε∆ω ≤ 0 a.e. in QT , ∂ω ω(0, x) = ̺0 − ̺ ≤ 0, = 0. ∂n ∂Ω (8.9) Test (8.9) by ω + = max{ω, 0} Z 1d + 2 kω kL2 (Ω) + ε |∇x ω + |2 dx Ω Z 2 dt 1 1 + 2 ≤− |ω | divx u dx ≤ kdivx ukL∞ (Ω) kω + k2L2 (Ω) 2 Ω 2 and thus d + 2 kω kL2 (Ω) ≤ kdivx ukL∞ (Ω) kω + k2L2 (Ω) . dt By the Gronwall inequality kω + (t, ·)kL2 (Ω) ≤ kω + (0, ·)kL2 (Ω) e Rt 0 kdivx ukL∞ (Ω) dτ =0 75 8.2. EXISTENCE FOR THE GALERKIN APPROXIMATION and thus ̺(t, x) − R(t) ≤ 0 a.e. in QT . Analogously, denoting r(t) = ̺e− Rt 0 kdivx u(τ )kL∞ (Ω) dτ , ω(t, x) = ̺(t, x) − r(t) ∂t ω + divx (ωu) − ε∆ω ≥ 0 a.e. in QT , ∂ω ω(0) = ̺0 − ̺ ≥ 0, = 0. ∂n ∂Ω (8.10) Testing by ω − implies kω − (t, ·)kL2 (Ω) = 0 and thus ̺(t, x) − r(t) ≥ 0 a.e. in QT . Whence (8.7). Step 4: (L2 bounds): a) Test (8.6) by ̺ Z Z d 2 2 k̺kL2 (Ω) + 2ε |∇x ̺| dx = − ̺2 divx u dx dt Ω Ω d =⇒ k̺k2L2 (Ω) ≤ kukW 1,∞ (Ω;R3 ) k̺k2L2 (Ω) . dt b) Test (8.6) by −∆̺ d dt Z Z 2 |∇x ̺| dx + 2ε |∆̺|2 dx Ω Ω Z Z = 2 ̺divx u∆̺ dx + 2 u · ∇x ̺∆̺ dx Ω ′ ≤ Ω ≤ CkukW 1,∞ (Ω;R3 ) k̺kW 1,2 (Ω) k∆̺kL2 (Ω) C kuk2W 1,∞ (Ω;R3 ) k̺k2W 1,2 (Ω) + εk∆̺k2L2 (Ω) . ε Therefore i.e. C d 2 2 3 k̺kW 1,2 (Ω) ≤ kukW 1,∞ (Ω;R ) + kukW 1,∞ (Ω;R3 ) k̺k2W 1,2 (Ω) , dt ε k̺kL∞ (0,t;W 1,2 (Ω)) ≤ k̺0,δ kW 1,2 (Ω) e C ε kukL∞ (0,t;W 1,2 (Ω;R3 )) +kuk2 ∞ L (0,t;W 1,2 (Ω;R3 )) t . 76 CHAPTER 8. EXISTENCE PROOF Further Z t k∆̺k2L2 (Ω) dτ ≤ k̺0 kW 1,2 (Ω) + ε 0 CkukL∞ (0,t;W 1,∞ (Ω;R3 )) k̺kL∞ (0,t;W 1,2 (Ω)) Z t 0 k∆̺kL2 (Ω) dτ which gives (8.8)2 . Similarly, testing (8.6) by ∂t ̺ Z t ε k∂t ̺k2L2 (Ω) dτ + k∇̺(t, ·)k2L2 (Ω;R3 ) 2 0 Z tZ ε ε 2 ≤ k∇̺0,δ kL2 (Ω;R3 ) + divx (̺u)∂t ̺ dx dt ≤ k∇̺0,δ k2L2 (Ω;R3 ) 2 2 0 Ω √ + C kukL∞ (0,T ;W 1,2 (Ω;R3 )) , k̺kL∞ (0,T ;W 1,2 (Ω;R3 )) , tk∂t ̺kL2 (0,T ;L2 (Ω)) , which yields (8.8)3 . Step 5: (Uniqueness) First, to get (8.8)4 , take u1 , u2 and subtract from the equation for ̺1 = ̺1 (u1 ) the equation for ̺2 = ̺2 (u2 ). It reads: ∂t (̺1 − ̺2 ) − ε∆(̺1 − ̺2 ) = −̺1 divx (u1 − u2 ) − ∇x ̺1 · (u1 − u2 ) −(̺1 − ̺2 )divx u2 − ∇x (̺1 − ̺2 )u2 . Test the above obtained equality by (̺1 − ̺2 ): d k̺1 − ̺2 k2L2 (Ω) + 2εk∇x (̺1 − ̺2 )k2L2 (Ω;R3 ) = Z dt − ̺1 divx (u1 − u2 ) − ∇x ̺1 · (u1 − u2 ) − (̺1 − ̺2 )divx u2 − ∇x (̺1 − ̺2 )u2 × Ω × (̺1 − ̺2 ) dx ≤ C(k̺1 kW 1,2 (Ω) k̺1 − ̺2 kL2 (Ω) ku1 − u2 kW 1,∞ (Ω;R3 ) + ku2 kW 1,∞ (Ω;R3 ) k̺1 − ̺2 k2L2 (Ω) ) and thus d k̺1 − ̺2 kL2 (Ω) dt ≤ Ck̺1 kW 1,2 (Ω) ku1 − u2 kW 1,∞ (Ω;R3 ) + Cku2 kW 1,∞ (Ω;R3 ) k̺1 − ̺2 kL2 (Ω) . 8.2. EXISTENCE FOR THE GALERKIN APPROXIMATION 77 Using Gronwall’s lemma k(̺1 − ̺2 )(t, ·)kL2 (Ω) ≤ Z t Rt k̺1 kW 1,2 (Ω) ku1 − u2 kW 1,∞ (Ω;R3 ) e τ Cku2 kW 1,∞ (Ω;R3 ) (s) ds dτ, C 0 which proves (8.8)4 and hence also the uniqueness. Remark 8.1 We can also show the renormalized continuity equation. Using the same method as in the proof of the validity of renormalized continuity equation, we have for any b sufficiently smooth, convex ∂t b(̺) + divx (b(̺)u) + (̺b′ (̺) − b(̺))divx u − ε∆b(̺) ≤ 0. Indeed, formally, multiplying the continuity equation by b′ (̺) ∂t b(̺) + divx (b(̺)u) + (̺b′ (̺) − b(̺))divx u − ε∆b(̺) = −εb′′ (̺)|∇x ̺|2 ≤ 0, where we used that ∆b(̺) = divx (b′ (̺)∇x ̺) = b′ (̺)∆̺ + b′′ (̺)|∇x ̺|2 . The details are similar as in the case with ε = 0, the only problematic term has a good sign. Note also that we can integrate the continuity equation over Ω (i.e. use as test function 1) Z Z d ̺(t, x) dx = 0 ⇒ ̺(t, x) dx = const. (in time). dt Ω Ω We now return to the full system with the Galerkin approximation for the velocity. We want to obtain a solution for the Galerkin approximation of ∂t (̺u) + divx (̺u ⊗ u) − µ∆u − (µ + λ)∇x divx u +∇x (̺γ + δ̺β ) + ε∇x ̺ · ∇x u = ̺f with ̺ being a solution to the (regularized) continuity equation with u. We shall apply the following version of the Schauder fixed point theorem (for the proof see e.g. [6]) Theorem 8.2 Let T : X → X be continuous and compact, X a Banach space. Let for any s ∈ [0, 1] the fixed points sT u = u be bounded. Then T possesses at least one fixed point. 78 CHAPTER 8. EXISTENCE PROOF We define the mapping T as follows. Take w ∈ C([0, T ]; Xn ), where Xn is the finite dimensional space spanned by the first n eigenvalues of −µ∆u − (µ + λ)∇x divx u with u|∂Ω = 0. We look for un , the Galerkin approximation of the momentum equation, i.e. for the solution to Z ∂t (̺(w)un ) · hi dx+ Z divx (̺(w)w ⊗ un ) · hi dx Z + µ ∇x un : ∇x hi dx+(µ + λ) divx un divx hi dx (8.11) Ω Ω Z Z γ β + (∇x ̺ (w) + δ∇x ̺ (w)) · hi dx + ε∇x ̺(w) · ∇x un · hi dx Ω Ω Z = ̺f · hi dx, un (0) = Pn u0 , i = 1, . . . , n. ΩZ Ω Ω Since for w ∈ C([0, T ]; Xn ) the solution to the regularized continuity equation is bounded away from zero, it is not difficult to see that there exists a solution to (8.11). Moreover, as the problem is linear, ∂t ̺ ∈ Lp ((0, T ) × Ω) for any p < ∞, by a standard energy method and Gronwall’s argument, the solution is unique. It is also possible to show that T is a continuous and compact mapping from C([0, T ]; Xn ) to itself. The main point is that we get an estimate of ∂t u, while in the spatial variable the compactness is just a consequence of the fact that Xn is finite dimensional. What remains is to show the boundedness of the possible fixed points. Take s ∈ [0, 1] and sT (un ) = un , i.e. T (un ) = un . s Then Z Z ∂t (̺un ) · un dx+ divx (̺un ⊗ un ) · un dx Ω Z Z + ε∇x ̺ · ∇x un · un dx + µ|∇x un |2 dx Ω Ω Z Z Z 2 γ β + (µ + λ)(divx un ) dx+s (∇x ̺ + δ∇x ̺ )un dx = s ̺f · un dx Ω Ω Ω Ω 8.2. EXISTENCE FOR THE GALERKIN APPROXIMATION 79 for s ∈ [0, 1]. Next, we have Z Z Z 1 1 2 ∂t ̺|un |2 dx, ∂t (̺un ) · un dx = ∂t ̺|un | dx + 2 2 Ω Ω Z Z Ω 1 divx (̺un ⊗ un ) · un dx = divx (̺un )|un |2 dx, 2 ΩZ ZΩ Z ε ε 2 ∇x ̺∇x |un | dx = − ∆̺|un |2 dx. ε∇x ̺ · ∇x un · un dx = 2 2 Ω Ω Ω R Summing these three integrals we get 21 ∂t Ω ̺|un |2 dx, due to the continuity equation. Further Z Z γ γ ̺un · ∇x ̺γ−1 dx ∇x ̺ · un dx = γ − 1 Ω Ω Z γ =− ̺γ−1 divx (̺u) dx γ−1 Ω Z Z 1 εγ γ ∂t ̺ dx − = ̺γ−1 ∆̺ dx γ−1 γ − 1 ZΩ Z Ω 1 ∂t ̺γ dx + εγ ̺γ−2 |∇̺|2 dx, = γ−1 Z ZΩ ZΩ 1 ∂t ̺β dx + εβ ̺β−2 |∇x ̺|2 dx. ∇x ̺β · un dx = β − 1 Ω Ω Ω Thus d s E (̺, un ) + µ dtZ δ +sεγ Ω Z Z |∇x un | dx + (µ + λ) (divx un )2 dx Ω Z Ω Z γ−2 2 β−2 2 ̺ |∇x ̺| dx + sεδβ ̺ |∇x ̺| dx ≤ ̺f · un dx, where Eδs (̺, un ) = 1 2 R Ω 2 Ω Ω ̺γ δ̺β ̺|un |2 + s γ−1 dx. As + s β−1 Z Z ̺f · un dx ≤ √̺√̺un · f dx Ω (8.12) Ω 1 1 ≤ k̺kL2 ∞ (0,T ;L1 (Ω)) kf kL∞ ((0,T )×Ω;R3 ) k̺|un |2 kL2 1 ((0,T )×Ω) , R we get the RL∞ (0, T ) control of the kinetic energy Ω ̺|un |2 dx and L1 (0, T ) control of Ω |∇x un |2 dx independently of s. As Xn is finite dimensional, 80 CHAPTER 8. EXISTENCE PROOF using (8.12) and (8.7), we see that ̺ is pointwisely controlled independently of s and thus, using once more (8.12), we see that kun kC([0,T ];Xn ) is also controlled independently of s. Therefore we can apply Theorem 8.2 to finish the proof of Theorem 8.1. Note that (8.4) follows from (8.12) integrating over (0, T ), setting s = 1 and ̺ := ̺n . 8.3 Estimates independent of n, limit passage n→∞ Recall that we have from the energy inequality (8.12) k̺n |un |2 kL∞ (0,T ;L1 (Ω)) ≤ C, k̺n kL∞ (0,T ;Lβ (Ω)) ≤ C, kun kL2 (0,T ;W 1,2 (Ω;R3 )) ≤ C, (8.13) β k̺n2 kL2 (0,T ;W 1,2 (Ω)) ≤ C. Note that 2 3 k̺n kL 35 β ((0,T )×Ω) ≤ k̺n kL5 ∞ (0,T ;Lβ (Ω)) k̺n kL5 β (0,T ;L3β (Ω)) ≤ C. (8.14) Next we test the continuity equation by ̺n . Z 1d 2 2 k̺n k2 + εk∇x ̺n k2 = − divx (̺n un )̺n dx 2 dt Ω Z Z 1 1 2 = un · ∇x ̺n dx = − ̺2 divx un dx. 2 Ω 2 Ω n Taking β ≥ 12 5 we therefore have k̺n kL2 (0,T ;W 1,2 (Ω)) ≤ C. (8.15) As we control 2β √ √ ̺n ̺n un in L∞ (0, T ; L β+1 (Ω; R3 )) √ √ (recall that ̺n is controlled in L∞ (0, T ; L2β (Ω)) and ̺n un is controlled in L∞ (0, T ; L2 (Ω; R3 ))), and ̺n un = 6β ̺n un in L2 (0, T ; L β+6 (Ω; R3 )) 8.3. ESTIMATES INDEPENDENT OF N , LIMIT PASSAGE N → ∞ 81 (recall that ̺n is controlled in L∞ (0, T ; Lβ (Ω)) and un in L2 (0, T ; L6 (Ω; R3 ))), 10β−6 then ̺n un is bounded in L 3(β+1) ((0, T ) × Ω; R3 ). Hence for β > 3 we control ̺n un in Ls̃ ((0, T ) × Ω; R3 ) for some s̃ > 2 and by virtue of Lemma 8.1 also ∇̺n in Ls̃ ((0, T ) × Ω; R3 ). Thus we know that divx (̺n un ) is bounded in Lq ((0, T ) × Ω) for some q > 1. Whence Lemma 8.1 implies estimates k∇2x ̺n kLq ((0,T )×Ω;R3×3 ) ≤ C, k∂t ̺n kLq ((0,T )×Ω) ≤ C. (8.16) We now recall our problem Z Z Z ∂t (̺n un ) · hi dx − ̺n (un ⊗ un ) : ∇x hi dx + µ ∇x un : ∇x hi dx Ω Ω Ω Z Z +(µ + λ) divx un divx hi dx− (̺γn + δ̺βn )divx hi dx Ω Z ZΩ +ε ∇x ̺n · ∇x un · hi dx = ̺n f · hi dx, (8.17) Ω Ω ∂t (̺n ) − ε∆̺n +divx (̺n un ) = 0, ∂̺n = 0. ∂n ∂Ω We have (for a chosen subsequence, denoted however again by the same index n) ∂t ̺n ⇀ ∂t ̺ in Lq ((0, T ) × Ω), ∇2x ̺n ⇀ ∇2x ̺ in Lq ((0, T ) × Ω; R3×3 ), ⇒ ∇x ̺n → ∇x ̺ in Lr ((0, T ) × Ω; R3 ) ∀r ≤ 2, ̺n ⇀∗ ̺ in L∞ (0, T ; Lβ (Ω)), 5 ̺n ⇀ ̺ in L 3 β ((0, T ) × Ω), 5 ̺n → ̺ in Lr ((0, T ) × Ω) ∀r < β, 3 un ⇀ u in L2 (0, T ; W 1,2 (Ω; R3 )), ⇒ ⇒ 10β−6 ̺n un ⇀ ̺u in L 3(β+1) ((0, T ) × Ω). Next we want to show that in fact ̺n un → ̺u strongly. To this aim, let us observe that for Pn the orthogonal projection from L2 (Ω; R3 ) to Xn we 82 CHAPTER 8. EXISTENCE PROOF have d dt Z Z Pn (̺n un ) · Φ dx = ̺n (un ⊗ un ) : ∇x Pn (Φ) dx Ω Z Z − µ ∇x un : ∇x Pn (Φ) dx − (µ + λ) divx un divx Pn (Φ) dx Ω Ω Z Z Z γ β + (̺n +δ̺n )divx Pn (Φ) dx−ε ∇x ̺n ·∇x un ·Pn (Φ) dx+ ̺n f ·Pn (Φ) dx, Ω Ω Ω Ω (8.18) t ∈ (0, T ) and Φ ∈ Cc∞ (Ω; R3 ). We now recall the properties of the projection Pn : Z Z Pn (u) · v dx = u · Pn (v) dx, ∀u, v ∈ L2 (Ω; R3 ), Ω Ω lim k(Pn − I)ukL2 (Ω;R3 ) = 0, n→∞ ∀u ∈ L2 (Ω; R3 ), kPn (u)kW k,2 (Ω;R3 ) ≤ CkukW k,2 (Ω;R3 ) , u∈ W01,2 (Ω; R3 ) ∩W k = 1, 2, k,2 (8.19) (Ω; R3 ), lim k(Pn − I)ukW 1,2 (Ω;R3 ) = 0, ∀u ∈ W01,2 (Ω; R3 ), n→∞ k(Pn − I)zk 2 6 L (Ω;R3 ) = 0, q> . lim sup n→∞ kzkW 1,q (Ω;R3 ) 5 z∈W 1,q (Ω;R3 );z6=0 It is an easy matter to observe that using (8.18) and (8.19) we have for some t > 1 k∂t (Pn (̺n un ))kLt (0,T ;W −2,2 (Ω;R3 )) ≤ C. Moreover, kPn (̺n un )kLq (0,T ;W 1,2 (Ω;R3 )) ≤ Ck̺n un kLq (0,T ;W 1,2 (Ω;R3 )) ≤ C, provided β > 15, i.e. 10β−6 3(β+1) Pn (̺n un ) → z, q > 1, > 3. Thus, by the Aubin–Lions lemma, strongly in Lq (0, T ; L2 (Ω; R3 ). It is not difficult to see, due to the fact that ̺n un converges to ̺u weakly, that z = ̺u. As k̺n un − ̺ukLq (0,T ;L2 (Ω;R3 )) ≤ k̺n un − Pn (̺n un )kLq (0,T ;L2 (Ω;R3 )) + kPn (̺n un ) − ̺ukLq (0,T ;L2 (Ω;R3 )) , 8.3. ESTIMATES INDEPENDENT OF N , LIMIT PASSAGE N → ∞ 83 we get due to (8.19)5 that ̺n un → ̺u in Lt (0, T ; L2 (Ω; R3 ). Whence in L3 ((0, T ) × Ω; R3 ). ̺n un → ̺u (8.20) Therefore we have for some s > 1 ̺n un ⊗ un ⇀ ̺u ⊗ u in Ls ((0, T ) × Ω; R3×3 ). (8.21) Finally, as ∇x ̺ is bounded in Ls1 ((0, T ) × Ω; R3 ), s1 > 3 we have that ∇x ̺n → ∇x ̺ in L3 ((0, T ) × Ω; R3 ) and ε∇x ̺n · ∇x un ⇀ ε∇x ̺ · ∇x u 6 in L 5 ((0, T ) × Ω; R3 ). Altogether we can pass to the limit in (8.17) to get − Z T Z (̺u) · Φ∂t ψ dx dt Z TZ Z TZ + ̺(u ⊗ u) : ∇x Φψ dx dt + µ∇x u : ∇x Φψ dx dt 0 ZΩ Z 0 Ω Z TZ T (̺γ + δ̺β )divx Φψ dx dt divx udivx Φψ dx dt − +(µ + λ) 0 Z ΩZ Z0 T ZΩ T ̺f · Φψ dx dt, ∇x ̺ · ∇x u · Φψ dx dt = +ε 0 0 Ω Ω 0 Ω (8.22) first for any Φ ∈ Lin{h1 , h2 , . . . } and ψ ∈ Cc∞ (0, T ), later due to density argument we could enlarge the space. As we do not need to specify the space now, we will not mention it explicitly. Similarly, after the limit passage in the continuity equation we get ∂t ̺ − ε∆̺ + divx (̺u) = 0, ∂̺ = 0, ∂n ∂Ω (8.23) satisfied a.e. and in the weak sense. Moreover, easily also u|∂Ω = 0. Finally, we may also pass to the limit in the energy inequality. As we control ∂t (̺u) in some negative space, then ̺u is continuous with values in some Lq (Ω; R3 ) 84 CHAPTER 8. EXISTENCE PROOF space. Hence, after a possible change on a set of zero measure, Z tZ Z 1 ̺γ ̺β 2 |∇x u|2 dx dτ (t, ·) dx + µ ̺|u| + +δ 2 γ − 1 β − 1 Ω Z tZ Z tZ 0 Ω 2 +(µ + λ) (divx u) dx dτ + εδβ ̺β−2 |∇x ̺|2 dx dτ 0 Ω 0 Ω Z tZ γ−2 2 ̺ |∇x ̺| dx dτ +εγ Z0 Ω Z tZ ̺γ ̺β 1 2 (0, ·) dx ̺|u| + +δ ̺f · u dx dτ + ≤ 2 γ−1 β−1 0 Ω Ω (8.24) for all t ∈ (0, T ]. 8.4 Estimates independent of ε, limit passage ε → 0+ Recall that from the energy inequality (8.24) we have k̺ε |uε |2 kL∞ (0,T ;L1 (Ω)) ≤ C, k̺ε kL∞ (0,T ;Lβ (Ω)) ≤ C, kuε kL2 (0,T ;W 1,2 (Ω;R3 )) ≤ C. (8.25) For β ≥ 4 we may test the continuity equation by ̺ε and get √ εk̺ε kL2 (0,T ;W 1,2 (Ω)) ≤ C. (8.26) However, at this point we need some better (and independent of ε) estimates of the pressure. We recall the properties of the Bogovskii operator (see (4.14)–(4.16) from Chapter 4). We test the momentum equation ∂t (̺ε uε ) + divx (̺ε uε ⊗ uε ) − µ∆uε − (µ + λ)∇x divx uε +∇x (̺γε + δ̺βε ) + ε∇x ̺ε · ∇x uε = ̺ε f by 1 ψ(t)B ̺ε − |Ω| Z ̺0,δ dx , Ω 8.4. ESTIMATES INDEPENDENT OF ε, LIMIT PASSAGE ε → 0+ with ψ ∈ Cc∞ (0, T ). Recall (1 ≤ p < ∞) p Z Z 1 ̺ε − ̺0,δ dx ≤ C(p, Ω) ̺pε dx. |Ω| Ω Ω p 85 Note further that Z 1 ̺0,δ dx ∂t ψ(t)B ̺ε − |Ω| Ω Z 1 ′ = ψ (t)B ̺ε − ̺0,δ dx + ψ(t)B(∂t ̺ε ). (8.27) |Ω| Ω Now, due to the continuity equation, ∂t ̺ε = −divx (̺ε uε ) + ε∆̺ε , therefore B (∂t ̺ε ) = −B(divx (̺ε uε )) + εB(∆̺ε ). We have Z where I2 T ψ 0 1 I1 = |Ω| Z Ω Z (̺γ+1 ε dt = 7 X Ij , j=1 T ψ 0 + δ̺β+1 ) dx ε Z Ω (̺γε + δ̺βε ) dx Z ̺0,δ dx dt, Ω Z 1 ̺0,δ dx dx dt (̺ε uε ) · ∂t ψB ̺ε − = − |Ω| Ω 0 Ω Z Z TZ 1 ̺0,δ dx + ψB (∂t ̺ε ) dx dt = − ̺ε uε · ∂t ψB ̺ε − |Ω| Ω 0 Ω Z TZ Z 1 ̺ε uε · ∂t ψB ̺ε − = − ̺0,δ dx dx dt |Ω| Ω 0 Ω Z TZ (̺ε uε · B(divx (̺ε uε )) − ε̺ε uε · B(∆̺ε )) ψ dx dt + Z T 0 Z Ω = I21 + I22 + I23 Z Z T Z 1 ̺0,δ dx dx dt, ψ ̺ε (uε ⊗ uε ) : ∇x B ̺ε − I3 = − |Ω| Ω 0 Ω 86 CHAPTER 8. EXISTENCE PROOF Z 1 ̺0,δ dx dx dt, I4 = ψ µ∇x uε : ∇x B ̺ε − |Ω| Ω 0 Ω Z T Z Z 1 ψ (µ + λ)divx uε ̺ε − ̺0,δ dx dx dt, I5 = |Ω| Ω 0 Ω Z T Z Z 1 ψ ε∇x ̺ε · ∇x uε · B ̺ε − I6 = ̺0,δ dx dx dt, |Ω| Ω 0 Ω Z Z T Z 1 ̺0,δ dx dx dt. I7 = − ψ ̺ε f · B ̺ε − |Ω| Ω 0 Ω Z Z T We estimate each term separately: |I1 | ≤ C(k̺ε kγL∞ (0,T ;Lγ (Ω)) + δk̺ε kβL∞ (0,T ;Lβ (Ω)) ) ≤ C(DATA), |I21 | ≤ C ≤ C ≤ C Z Z Z T 0 T 0 T 0 Z 1 |̺ε ||uε | B ̺ε − ̺0,δ dx dx dt |Ω| Ω Ω Z 1 ̺0,δ dx dt k̺ε kLβ (Ω) kuε kL6 (Ω;R3 ) B ̺ε − 6β 5β−6 |Ω| Ω L (Ω;R3 ) Z k̺ε kLβ (Ω) kuε kL6 (Ω;R3 ) k̺ε kLβ (Ω) dt ≤ C(DAT A) if β ≥ |I22 | ≤C ≤C Z ≤ C Z if β ≥ 3, |I23 | ≤ C T 0 Z T 0 T 0 ≤C Z Z Z T 0 Z Ω 12 , 7 ̺ε uε · B(divx (̺ε uε )) dx dt T 0 Ω Z Ω k̺ε kL3 (Ω) kuε kL6 (Ω;R3 ) k̺ε uε kL2 (Ω;R3 ) k̺ε k2L3 (Ω) kuε k2L6 (Ω;R3 ) dx dt ≤ C(DATA) |̺ε uε |B(ε∆̺ε ) dx dt εk∇̺ε kL2 (Ω;R3 ) k̺ε kL3 (Ω) kuε kL6 (Ω;R3 ) ≤ C(DAT A) 8.4. ESTIMATES INDEPENDENT OF ε, LIMIT PASSAGE ε → 0+ if β ≥ 3, |I3 | ≤ C ≤ C Z Z T 0 T 0 87 Z 1 2 ̺0,δ dx dx dt ̺ε |uε | ∇x B ̺ε − |Ω| Ω Ω Z k̺ε k2L3 (Ω) kuε k2L6 (Ω;R3 ) dx dt ≤ C(DATA) if β ≥ 3, |I4 | + |I5 | ≤ Z Z T 1 ̺0,δ dx C k∇x uε kL2 (Ω;R3×3 ) ∇x B ̺ε − dt 2 |Ω| Ω 0 L (Ω;R3×3 ) Z T ≤C k∇x uε kL2 (Ω;R3×3 ) k̺ε kL2 (Ω) dt, 0 |I6 | ≤ C Z T 0 Z ≤C εk∇x ̺ε kL2 (Ω;R3 ) k∇x uε kL2 (Ω;R3×3 ) × Z 1 × ̺0,δ dx dt B ̺ε − |Ω| ∞ Ω L (Ω;R3 ) T 0 εk∇x ̺ε kL2 (Ω;R3 ) k∇x uε kL2 (Ω;R3×3 ) k̺ε kLβ (Ω) dt ≤ C(DAT A) if β > 3, |I7 | ≤ C Z T 0 Z 1 ̺0,δ dx dt k̺ε kLβ (Ω) β β−1 B ̺ε − |Ω| Ω L (Ω;R3 ) Z T ≤C k̺ε k2Lβ (Ω) dt 0 if β ≥ 32 . Therefore, Z T 0 Z Ω ψ(̺γ+1 + δ̺β+1 ) dx dt ≤ C(DATA), ε ε provided β > 3. ψ ∈ Cc∞ (I), 88 CHAPTER 8. EXISTENCE PROOF Hence 1 k̺ε kLγ+1 ((0,T );Lγ+1 (Ω)) + δ β+1 k̺ε kLβ+1 ((0,T );Lβ+1 (Ω)) ≤ C. loc loc (8.28) As in the weak sequential stability one can show that (cf. Theorem 7.2) 2β ̺ε uε ⇀ ̺u in Cweak (0, T ; L β+1 (Ω; R3 )). Moreover ̺ε → ̺ in Cweak (0, T ; Lβ (Ω)). Writing the continuity equation (8.23) in the weak form, Z TZ Z TZ ε∇x ̺ε · ∇x Φ dx dt ̺ε ∂t Φ dx dt + − 0 Ω 0 Ω Z TZ ̺ε uε · ∇x Φ dx dt = 0 ∀Φ ∈ Cc∞ ((0, T ) × Ω), − 0 thus for ε → 0 − Ω Z T 0 Z Ω ̺∂t Φ dx dt − Z T 0 Z Ω ̺u · ∇x Φ dx dt = 0 (8.29) for all Φ ∈ W 1,2 ((0, T ) × Ω) compactly supported in (0, T ). Whence we recover the weak formulation of the continuity equation. Next we consider the momentum equation. Since Z TZ |∇x ̺ε · ∇x uε | dx dt ε 0 Ω √ √ ≤ ε εk∇x ̺ε kL2 (0,T ;L2 (Ω;R3 )) k∇uε kL2 ((0,T )×Ω;R3×3 ) , and exactly as in Chapter 6 (the proof of the weak sequential stability for γ ≥ 5), due to the Div-Curl Lemma ̺ε uε ⊗ uε ⇀ ̺u ⊗ u in Lq ((0, T ) × Ω; R3×3 ) for some q > 1, we recover after the limit passage ε → 0+ Z TZ Z TZ ̺(u ⊗ u) : ∇x Φ dx dt ̺u · ∂t Φ dx dt − − 0 Ω Z Z T Z0 Ω T Z divx u divx Φ dx dt µ∇x u : ∇x Φ dx dt + (µ + λ) + 0 Ω 0 Ω Z TZ Z TZ γ β − (̺ + δ̺ )divx Φ dx dt = ̺f · Φ dx dt, 0 Ω 0 Ω (8.30) 8.4. ESTIMATES INDEPENDENT OF ε, LIMIT PASSAGE ε → 0+ 89 for any Φ ∈ Cc∞ ((0, T ) × Ω; R3 ). Hence, by density argument, also for Φ 6β bounded with ∂t Φ ∈ L2 (0, T ; L 5β−1 (Ω; R3 )), ∇Φ ∈ L2 ((0, T ) × Ω; R3×3 ) and divx Φ ∈ Lβ+1 ((0, T ) × Ω), with compact support in time. The last task is to show that ̺γ + δ̺β = ̺γ + δ̺β , i.e. the strong convergence of the density. 8.4.1 Strong convergence of the density First recall that we have the following renormalized formulation of the continuity equation ∂t (b(̺ε )) + divx (b(̺ε )uε ) + (̺ε b′ (̺ε ) − b(̺ε ))divx uε − ε∆b(̺ε ) = −εb′′ (̺ε )|∇̺ε |2 ≤ 0 (8.31) with b sufficiently smooth and convex. Due to the fact β ≫ 1, we also have in the limit (we can prove it directly from the weak formulation of the continuity equation for the limit functions, see Lemma 7.3) ∂t (b(̺)) + divx (b(̺)u) + (̺b′ (̺) − b(̺))divx u = 0 (8.32) for b sufficiently smooth. We now proceed as in the weak sequential compactness part. The aim is to show the effective viscous flux identity ̺γ+1 + δ̺β+1 − (2µ + λ)̺divx u = ̺γ ̺ + δ̺β ̺ − (2µ + λ)̺divx u a.e. in Ω. (8.33) To show (8.33) we proceed exactly as in Chapter 6. One difference is that for ϕ ε = ∇x ∆−1 [̺ε 1Ω ] we have ∂tϕ ε = ∇x ∆−1 [∂t ̺ε 1Ω ] = −∇x ∆−1 divx (̺ε uε ) + ε∇x (̺ε 1Ω ) . | {z } →0 Next, unlike Chapter 3, ∆−1 represents here the inverse of the Laplacean on R3 , specifically, iξj −1 ∂xj ∆ [v] = Fξ→x Fx→ξ [v] . |ξ|2 Finally, the term ε Z T 0 Z Ω ∇x ̺ε · ∇x uε · ϕ ε dx dt → 0 90 CHAPTER 8. EXISTENCE PROOF for ε → 0+ . The rest is the same, hence we obtain the effective viscous flux identity. We therefore use in the renormalized continuity equation b(̺) = ̺ ln ̺. Note that b′′ (̺) = (ln ̺ + 1)′ = ̺1 > 0, i.e. it is a convex function. We integrate it over Ω and get Z Z d ̺ ln ̺ dx + ̺divx u = 0 in D′ ((0, T )). dt Ω Ω But ̺ ln ̺ ∈ C([0, T ]) and thus Z Z Z tZ ̺divx u dx dτ = 0. (̺ ln ̺)(t, ·) dx − (̺ ln ̺)(0, ·) dx + Ω 0 Ω Ω Further, for ε > 0 we have ∂t (̺ε ln ̺ε ) + divx (̺ε ln ̺ε uε ) + ̺ε divx uε − ε∆(̺ε ln ̺ε ) ≤ 0. Integrating it over Ω (recall that ∂n ̺ε |∂Ω = 0) Z Z d ̺ε ln ̺ε dx + ̺ε divx uε dx ≤ 0 dt Ω Ω and thus Z Z Z tZ (̺ε ln ̺ε )(t, ·) dx − (̺ε ln ̺ε )(0, ·) dx + ̺ε divx uε dx dτ ≤ 0. Ω Ω 0 Ω Passing with ε → 0+ Z Z tZ Z ̺divx u dx dτ ≤ 0. (̺ ln ̺)(t, ·) dx − (̺ ln ̺)(0, ·) dx + 0 Ω Ω Ω Therefore Z Z tZ (̺ ln ̺)(t, ·) − (̺ ln ̺)(t, ·) dx ≤ (̺divx u − ̺divx u) dx dτ Ω 0 Ω Z t Z 1 dx dτ. ̺γ ̺ − ̺γ+1 + δ ̺β ̺ − ̺β+1 = 2µ + λ 0 Ω As ̺γ+1 − ̺γ ̺ = lim+ (̺γ+1 − ̺γε ̺) = lim+ (̺γε − ̺γ )(̺ε − ̺) ≥ 0, ε ε→0 ε→0 8.5. ESTIMATES INDEPENDENT OF δ, LIMIT PASSAGE δ → 0+ 91 we have ̺γ ̺ ≤ ̺γ+1 and thus Z (̺ ln ̺)(t, ·) − ̺ ln ̺(0, ·) dx ≤ 0. Ω This yields as before (lower weak semicontinuity of convex function, see Chapter 6) ̺ ln ̺ = ̺ ln ̺ and we arrive at the same point as before, i.e. we prove the strong convergence of the density. Hence we get the weak formulation of the momentum equation Z TZ Z TZ ̺(u ⊗ u) : ∇x Φ dx dt ̺u · ∂t Φ dx dt − − 0 Ω Z Z T Z0 Ω T Z (8.34) divx u divx Φ dx dt µ∇x u : ∇x Φ dx dt + (µ + λ) + 0 Z0 T ZΩ ZΩ T Z ̺f · Φ dx dt, (̺γ + δ̺β ) divx Φ dx dt = − 0 Ω 0 Ω and the energy inequality Z 1 ̺γ ̺β 2 (t, ·) dx ̺|u| + +δ γ−1 βZ − 1Z Z t ZΩ 2 t +µ |∇x u|2 dx dτ + (µ + λ) (divx u)2 dx dτ 0 Ω Z Z tZ 0 Ω ̺γ ̺β 1 2 ̺|u| + +δ (0, ·) dx ̺f · u dx dτ + ≤ 2 γ−1 β−1 0 Ω Ω (8.35) for all t ∈ (0, T ). 8.5 Estimates independent of δ, limit passage δ → 0+ We have as before k̺δ |uδ |2 kL∞ (0,T ;L1 (Ω)) ≤ C, 1 k̺δ kL∞ (0,T ;Lγ (Ω)) + δ β k̺δ kL∞ (0,T ;Lβ (Ω)) ≤ C, kuδ kL2 (0,T ;W 1,2 (Ω;R3 )) ≤ C, (8.36) 92 CHAPTER 8. EXISTENCE PROOF and ̺δ → ̺ in Cweak (0, T ; Lγ (Ω)), 2γ ̺δ uδ → ̺u in Cweak (0, T ; L γ+1 (Ω; R3 )). (8.37) We need to estimate the pressure in a better space than just L1 ((0, T )×Ω). To this aim, we apply similar type of improved pressure estimates as in the previous limit passage in Section 8.4. However, we have to employ a slightly different test function, namely Z 1 Θ Θ ψ(t)B ̺δ − ̺ dx , |Ω| Ω δ with ψ ∈ Cc∞ (0, T ) and Θ > 0. Recall that p Z Z Θ 1 pΘ Θ ̺δ − ̺ dx ≤ C(p, Ω) ̺δ dx |Ω| Ω δ Ω p for any 1 ≤ p < ∞, and Z 1 Θ ̺ dx − ∂t ψ(t)B |Ω| Ω δ Z Z 1 1 ′ Θ Θ Θ Θ = ψ (t)B ̺δ − ̺ dx + ψ(t)B ∂t ̺δ − ∂t ̺δ dx . (8.38) |Ω| Ω δ |Ω| Ω ̺Θ δ Due to the renormalized continuity equation, Θ Θ ∂ t ̺Θ δ = −divx (̺δ uδ ) − (Θ − 1)̺δ divx u, therefore Z 1 Θ Θ ∂t ̺δ dx B ∂ t ̺δ − |Ω| Ω = Thus −B(divx (̺Θ δ uδ )) Z T ψ 0 Z − (Θ − 1)B Ω ̺Θ δ divx uδ 1 − |Ω| (̺γ+Θ + δ̺β+Θ ) dx dt = δ δ 6 X j=1 Z Ij , Ω ̺Θ δ divx uδ dx . 8.5. ESTIMATES INDEPENDENT OF δ, LIMIT PASSAGE δ → 0+ where 1 I1 = |Ω| I2 Z T ψ 0 Z Ω (̺γδ δ̺βδ ) dx + Z Ω ̺Θ dx dt, δ Z 1 Θ ̺ dx dx dt (̺δ uδ ) · ∂t ψB = − − |Ω| Ω δ 0 Ω " Z TZ Z 1 Θ Θ ̺δ uδ · ∂t ψB ̺δ − = − ̺ dx |Ω| Ω δ 0 Ω # Z 1 + ψB ∂t ̺Θ ∂ t ̺Θ dx dt δ − δ dx |Ω| Ω Z Z TZ 1 Θ Θ ̺ dx dx dt ̺δ uδ · ∂t ψB ̺δ − = − |Ω| Ω δ 0 Ω Z TZ ̺δ uδ · B(divx (̺δ uδ )) + (Θ − 1)̺δ uδ · B ̺Θ + δ divx uδ Z Z T 0 Ω Z 1 − |Ω| ̺Θ δ divx uδ Ω = I21 + I22 + I23 I3 = − Z I4 = I5 = T 0 Z Z dx ! ̺Θ δ ψ dx dt Z 1 Θ Θ ̺ dx dx dt, ψ ̺δ (uδ ⊗ uδ ) : ∇x B ̺δ − |Ω| Ω δ Ω Z T 0 Z 1 Θ Θ ̺ dx dx dt, ψ µ∇x uδ : ∇x B ̺δ − |Ω| Ω δ Ω Z T ψ 0 I6 = − Z Z (µ + λ)divx uδ Ω T ψ 0 Z Ω ̺δ f · B ̺Θ δ ̺Θ δ 1 − |Ω| 1 − |Ω| Z Z Ω Ω ̺Θ δ ̺Θ δ dx dx dt. dx dx dt, We estimate each term separately: β+Θ |I1 | ≤ C(k̺δ kγ+Θ L∞ (0,T ;Lγ (Ω)) + δk̺δ kL∞ (0,T ;Lβ (Ω)) ) ≤ C(DATA), 93 94 CHAPTER 8. EXISTENCE PROOF provided Θ ≤ γ, |I21 | ≤ C ≤ C ≤ C Z Z Z T 0 T 0 T Z 1 Θ Θ |̺δ ||uδ | B ̺δ − ̺δ dx dx dt |Ω| Ω Ω Z 1 Θ Θ ̺ dx dt B ̺ − k̺δ kLγ (Ω) kuδ kL6 (Ω;R3 ) δ δ 6γ 5γ−6 |Ω| Ω L (Ω;R3 ) Z k̺δ kLγ (Ω) kuδ kL6 (Ω;R3 ) k̺δ kΘ Lγ (Ω) dt ≤ C(DAT A), 0 if Θ ≤ 65 γ − 1, |I22 | ≤C ≤C if Θ ≤ |I23 | 2 γ 3 ≤ C ≤ C T 0 Z T T 0 Z Ω ̺δ uδ B(divx (̺δ uδ )) dx dt T k̺δ kL 23 (1+Θ) (Ω) kuδ kL6 (Ω;R3 ) k̺δ uδ k L 0 Z − 1, Z Z ≤C Z T 0 Z Z Ω k̺δ k1+Θ 3 (1+Θ) L2 (Ω) 6(1+Θ) 1+5Θ (Ω;R3 ) dt kuδ k2L6 (Ω;R3 ) dx dt ≤ C(DATA) Z 1 Θ ̺δ |uδ |B ̺δ divx uδ − ̺Θ div u dx dx dt x δ δ Ω| Ω Ω k̺δ kLγ (Ω) kuδ kL6 (Ω;R3 ) × Z 1 Θ ̺Θ div u dx × B ̺δ divx uδ − dt 6γ 5γ−6 x δ δ |Ω| Ω L (Ω;R3 ) Z T ≤ C k̺δ kLγ (Ω) kuδ kL6 (Ω;R3 ) kdivx uδ kL2 (Ω) k̺δ kΘ 3γΘ ≤ C(DAT A) 0 L 2γ−3 (Ω) 0 if Θ ≤ 23 γ − 1 and γ ≤ 6, |I3 | ≤ C ≤ C Z Z T 0 T 0 Z Z 1 Θ Θ ̺δ |uδ | ∇x B ̺δ − ̺δ dx dx dt |Ω| Ω Ω 2 k̺δ kLγ (Ω) kuδ k2L6 (Ω;R3 ) k̺δ kΘ 3γΘ L 2γ−3 (Ω) dx dt ≤ C(DATA) 8.5. ESTIMATES INDEPENDENT OF δ, LIMIT PASSAGE δ → 0+ 95 if Θ ≤ 32 γ − 1, |I4 | + |I5 | ≤ Z T Z 1 Θ Θ C k∇x uδ kL2 (Ω;R3×3 ) ̺δ dx dt ∇x B ̺δ − |Ω| 2 0 Ω L (Ω;R3×3 ) Z T ≤C k∇x uδ kL2 (Ω;R3×3 ) k̺δ kL2Θ (Ω) dt ≤ C(DAT A), 0 if Θ ≤ γ2 , |I6 | ≤ C Z T 0 Z 1 Θ Θ dt ̺δ dx k̺δ kLγ (Ω) B ̺δ − γ γ−1 |Ω| Ω (Ω;R3 ) L Z T k̺δ k1+Θ ≤C Lγ (Ω) dt 0 if Θ ≤ 34 Θ − 1. Therefore, Z T 0 Z Ω ψ(̺γ+Θ + δ̺β+Θ ) dx dt ≤ C(DATA), δ δ ψ ∈ Cc∞ (I), provided (we consider only γ ≤ 6, the other case is easier and can be treated with analogy to Chapter 6) 2 Θ ≤ γ − 1. 3 Hence k̺δ kLγ+Θ ((0,T );Lγ+Θ (Ω)) + δk̺δ kLβ+Θ ((0,T );Lβ+Θ (Ω)) ≤ C. loc loc (8.39) As above we can pass to the limit in the weak formulation of the continuity equation to get Z TZ Z TZ − ̺∂t Φ dx dt − ̺u · ∇x Φ dx dt = 0 (8.40) 0 Ω 0 Ω for all Φ ∈ W 1,6 ((0, T ) × Ω). Moreover, ̺δ uδ ⊗ uδ ⇀ ̺u ⊗ u in Lq ((0, T ) × Ω; R3×3 ) for some q > 1 96 CHAPTER 8. EXISTENCE PROOF + and we may pass to the limit R T δR → β0 in the weak formulation of the momentum equation (note that δ 0 Ω ̺δ divx Φ dx dt → 0) + Z − T 0 Z T Z T Z ̺(u ⊗ u) : ∇x Φ dx dt Z TZ divx u divx Φ dx dt µ∇x u : ∇x Φ dx dt + (µ + λ) 0 Ω Ω Z TZ Z TZ γ − ̺f · Φ dx dt ̺ divx Φ dx dt = Z0 Ω 0 ̺u · ∂t Φ dx dt − Z 0 Ω Ω 0 (8.41) Ω for all Φ ∈ Cc∞ ((0, T ) × Ω; R3 ). To finish the proof it remains to show that ̺γ = ̺γ . Recall that, due to restriction coming from above, we consider γ > 32 . 8.5.1 Strong convergence of the density We will follow a similar strategy as before, i.e. we show • effective viscous flux identity • validity of the renormalized continuity equation • strong convergence of the density Recall that we control k̺δ kLγ+θ (I×Ω) ≤ C, where θ = 23 γ − 1. Then 35 γ − 1 = 2 for γ = 95 , i.e. for γ < 95 there is an additional difficulty: for the limit (̺, u) we have not guaranteed the renormalized continuity equation, we will have to verify it differently. We denote for z ∈ [0, 1], z ∈ (1, 2) concave for z ∈ [1, 3], T (z) = 2 for z ≥ 3 with T (·) ∈ C ∞ (R+ 0 ), and Tk (z) = kT z k , k ∈ N. 8.5. ESTIMATES INDEPENDENT OF δ, LIMIT PASSAGE δ → 0+ 97 We aim at showing ̺γ Tk (̺) − (2µ + λ)Tk (̺)divx u = ̺γ Tk (̺) − (2µ + λ)Tk (̺)divx u (8.42) a.e. in (0, T ) × Ω for all k ∈ N. The proof is based on a similar idea as before; we use a clever test function for approximated momentum equation for δ > 0, then for the limit problem; finally we pass to the limit δ → 0+ , using certain tools from the compensated compactness theory. Recall that we have for δ > 0 the renormalized continuity equation in the form ∂t (Tk (̺δ )) + divx (Tk (̺δ )uδ ) + (̺δ Tk′ (̺δ ) − Tk (̺δ ))divx uδ = 0, however, for the limit we only have (in fact, for γ ≥ however, we are mainly interested in low γ’s) 9 5 (8.43) the situation is better, ∂t (Tk (̺)) + divx (Tk (̺)u) + (̺Tk′ (̺) − Tk (̺))divx u = 0. (8.44) We use as the test function in the approximated momentum equation (understood in the weak sense) γ β ∂t (̺δ uδ ) + divx (̺δ uδ ⊗ uδ ) − µ∆uδ − (µ + λ)∇x divx uδ + ∇x ̺δ + δ̺δ = ̺δ f the function ϕδ = φ∇x ∆−1 (1Ω Tk (̺δ )), k ∈ N and for the limit equation (again, understood in the weak sense) ∂t (̺u) + divx (̺u ⊗ u) − µ∆u − (µ + λ)∇x divx u + ∇x ̺γ = ̺f the test function ϕ = φ∇x ∆−1 (1Ω Tk (̺)), k ∈ N. Here, ∆−1 represents as in the previous section the inverse of the Laplacean on R3 , i.e. iξj −1 Fx→ξ [v] , ∂xj ∆ [v] = Fξ→x |ξ|2 and φ ∈ Cc∞ ((0, T ) × Ω). Note that for 1 ≤ p < 3 we have k∇x ∆−1 [v]k 3p L 3−p (Ω;R3 ) ≤ CkvkLp (Ω) 98 CHAPTER 8. EXISTENCE PROOF and for p > 3 k∇x ∆−1 [v]kC(Ω;R3 ) ≤ CkvkLp (Ω) . Step 1: As ̺δ → ̺ in Cweak ([0, T ]; Lγ (Ω)), we have, in accordance with the standard Sobolev embedding relation W 1,p (Ω) ֒→֒→ C(Ω), p > 3, ∇x ∆−1 [1Ω Tk (̺δ )] → ∇x ∆−1 [1Ω Tk (̺)] in C([0, T ] × Ω). Now for φ ∈ Cc∞ ((0, T ) × Ω) Z τZ lim φp(̺δ )Tk (̺δ ) + p(̺δ )∇x φ · ∇x ∆−1 [1Ω Tk (̺δ )] dx dt δ→0 0 − lim δ→0 Z (8.45) Ω τ 0 Z Ω φ µ∇x uδ : ∇2x ∆−1 [1Ω Tk (̺δ )] + (λ + µ)divx uδ Tk (̺δ ) dx dt − lim δ→0 Z τ 0 Z µ∇x uδ · ∇φ · ∇x ∆−1 [1Ω Tk (̺δ )] Ω +(λ + µ)divx uδ ∇x φ · ∇x ∆−1 [1Ω Tk (̺δ )] dx dt Z τZ φp(̺)Tk (̺) − p(̺)∇x φ · ∇x ∆−1 [1Ω Tk (̺)] dx dt = − Z 0 Z τ 0 Ω Ω 2 −1 φ µ∇x u : ∇x ∆ [1Ω Tk (̺)] + (λ + µ)divx uTk (̺) dx dt − + lim δ→0 Z τ 0 − Z Z τ 0 Z Ω µ∇x u · ∇φ · ∇x ∆−1 [1Ω Tk (̺)] +(λ + µ)divx u∇x φ · ∇x ∆−1 [1Ω Tk (̺)] dx dt Z φ̺δ uδ · ∇x ∆−1 [divx (Tk (̺δ )uδ ) + (̺δ Tk′ (̺δ ) − Tk (̺δ ))divx uδ ] Ω −̺δ (uδ ⊗ uδ ) : ∇x φ∇x ∆−1 [1Ω Tk (̺δ )] τ 0 dx dt Z φ̺u · ∇x ∆−1 [divx (Tk (̺)u) + (̺Tk′ (̺) − Tk (̺))divx u] Ω −1 −̺(u ⊗ u) : ∇x φ∇x ∆ [1Ω Tk (̺)] dx dt 8.5. ESTIMATES INDEPENDENT OF δ, LIMIT PASSAGE δ → 0+ − lim δ→0 + Z Z Z τ 0 τ 0 Z ∂t φ̺δ uδ · ∇x ∆−1 (Tk (̺δ )) dx dt Ω Ω 99 ∂t φ̺u · ∇x ∆−1 (Tk (̺)) dx dt. Step 2: We have Z φ∇x uδ : Ω ∇2x ∆−1 [1Ω Tk (̺δ )] = Z X 3 − = − Ω i,j=1 φdivx uδ Tk (̺δ ) dx + Ω Z X 3 Ω i,j=1 ∂xj uiδ [∂xi ∆−1 ∂xj ][1Ω Tk (̺δ )] dx ∂xj φuiδ [∂xi ∆−1 ∂xj ][1Ω Tk (̺δ )] dx Ω i,j=1 Z φ 3 X ∂xj (φuiδ )[∂xi ∆−1 ∂xj ][1Ω Tk (̺δ )] dx Ω i,j=1 Z X 3 dx = Z Z Ω ∇x φ · uδ Tk (̺δ ) dx ∂xj φuiδ [∂xi ∆−1 ∂xj ][1Ω Tk (̺δ )] dx. Consequently, going back to (8.45) and dropping the compact terms, we obtain Z τZ lim φ p(̺δ )Tk (̺δ ) − (λ + 2µ)divx uδ Tk (̺δ ) dx dt (8.46) δ→0 0 − Ω Z τ 0 Z Ω = lim δ→0 φ p(̺)Tk (̺) − (λ + 2µ)divx uTk (̺) dx dt Z τ 0 Z Ω φ ̺δ uδ · ∇x ∆−1 [divx (Tk (̺δ )uδ )] −1 − Z −̺δ (uδ ⊗ uδ ) : ∇x ∆ ∇x [1Ω Tk (̺δ )] dx dt τ 0 Z φ̺u · ∇x ∆−1 [divx (Tk (̺)u)] − ̺(u ⊗ u) : ∇x ∆−1 ∇x [1Ω Tk (̺)] dx dt. Ω 100 CHAPTER 8. EXISTENCE PROOF Step 3: Our goal is to show that the right-hand side of (8.46) vanishes. We write Z h i −1 −1 φ ̺δ uδ · ∇x ∆ [1Ω divx (Tk (̺δ )uδ )] − ̺δ (uδ ⊗ uδ ) : ∇x ∆ ∇x [1Ω Tk (̺δ )] dx Ω = Z Ω h i φuδ · Tk (̺δ )∇x ∆−1 [divx (1Ω ̺δ uδ )]−̺δ uδ ·∇x ∆−1 ∇x [1Ω Tk (̺δ )] dx+l.o.t., where l.o.t. denotes lower order terms (with derivatives on φ). As in Chapter 6, we consider the bilinear form [v, w] = 3 X i,j=1 i j i j v Ri,j [w ] − w Ri,j [v ] , Ri,j = ∂xi ∆−1 ∂xj , writing 3 X i j i j v Ri,j [w ] − w Ri,j [v ] i,j=1 = 3 X (v i − Ri,j [v j ])Ri,j [wj ] − (wi − Ri,j [wj ])Ri,j [v j ] i,j=1 = U · V − W · Z, where 3 3 X X i j i U = (v − Ri,j [v ]), W = (wi − Ri,j [wj ]), divx U = divx W = 0, i j=1 j=1 and V i = ∂ xi 3 X j=1 ∆−1 ∂xj wj ! , Z i = ∂ xi 3 X j=1 ∆−1 ∂xj v j ! , i = 1, 2, 3. Therefore we may apply the Div-Curl lemma (Lemma 6.1) and using Tk (̺δ ) → Tk (̺δ ) in Cweak ([0, T ]; Lq (Ω)), 1 ≤ q < ∞, ̺δ uδ → ̺u in Cweak ([0, T ]; L2γ/(γ+1) (Ω; R3 )) 8.5. ESTIMATES INDEPENDENT OF δ, LIMIT PASSAGE δ → 0+ 101 we conclude that Tk (̺δ )(t, ·)∇x ∆−1 [1Ω divx (̺δ uδ )(t, ·)] − (̺δ uδ )(t, ·) · ∇x ∆−1 ∇x [1Ω Tk (̺δ )(t, ·)] (8.47) → Tk (̺)(t, ·)∇x ∆−1 [1Ω divx (̺u)(t, ·)] − (̺u)(t, ·) · ∇x ∆−1 ∇x [1Ω Tk (̺)(t, ·)] weakly in Ls (Ω; R3 ) for all t ∈ [0, T ], with 2γ 6 3 > since γ > . γ+1 5 2 Thus the convergence in (8.47) takes place in the space s< Lq (0, T ; W −1,2 (Ω)) for any 1 ≤ q < ∞; going back to (8.46), we have lim δ→0 Z τ 0 = Z Z Ω 0 τ φ p(̺δ )Tk (̺δ ) − (λ + 2µ)divx uδ Tk (̺δ ) dx dt Z Ω (8.48) φ p(̺) Tk (̺) − (λ + 2µ)divx uTk (̺) dx dt. Therefore, localizing we get as in Chapter 6 the desired form of the effective viscous flux identity (8.42). Next we want to verify that for γ > 32 we have the renormalized continuity equation (with b(̺) = Tk (̺)) fulfilled. For γ ≥ 59 we get this immediately, as ̺ belongs to L2 ((0, T ) × Ω). But for γ < 95 additional work is required. We introduce the quantity oscillation defect measure oscq (̺δ − ̺) := sup lim sup kTk (̺δ ) − Tk (̺)kLq ((0,T )×Ω) . k≥1 δ→0+ Below, we shall show (i) oscγ+1 (̺δ − ̺) < ∞ (ii) Z T Z lim sup |Tk (̺δ ) − Tk (̺)|γ+1 dx dt + δ→0Z 0 Ω T Z ≤ (̺γ Tk (̺) − ̺γ Tk (̺)) dx dt, 0 Ω (8.49) 102 CHAPTER 8. EXISTENCE PROOF (iii) if oscq (̺δ − ̺) < ∞ for some q > 2, the limit functions (̺, u) fulfill the renormalized continuity equation (with b(̺) a bounded, smooth function) and hence, by density argument, also for less regular functions. Lemma 8.3 We have (ii), i.e. (8.49), and (i). Proof: We have Z T 0 Z Z Ω T Z ̺γ Tk (̺) − ̺γ Tk (̺) dx dt = lim+ (̺γδ − ̺γ )(Tk (̺δ ) − Tk (̺)) dx dt δ→0 0 Ω Z TZ + (̺γ − ̺γ )(Tk (̺) − Tk (̺)) dx dt. 0 Ω However, the second term is nonnegative, as ̺ 7→ ̺γ is convex, ̺ 7→ Tk (̺) is concave, i.e. ̺γ ≤ ̺γ and Tk (̺) ≥ Tk (̺). Next, as |Tk (t) − Tk (s)| ≤ |t − s|, (t − s)γ ≤ (tγ − sγ ), t, s ≥ 0 t ≥ s ≥ 0, we get (Tk (t) − Tk (s))(tγ − sγ ) ≥ |Tk (t) − Tk (s)|γ+1 , Hence Z T Z lim sup |Tk (̺δ ) − Tk (̺)|γ+1 dx dt δ→0+ 0 Ω Z TZ ≤ (̺γ Tk (̺) − ̺γ Tk (̺)) dx dt 0 Ω t, s ≥ 0. 8.5. ESTIMATES INDEPENDENT OF δ, LIMIT PASSAGE δ → 0+ 103 which proves (ii). Using now (8.42), we have the identity Z TZ γ γ ̺ Tk (̺)−̺ Tk (̺) dx dt 0 Ω Z TZ = (2µ + λ) lim+ divx uδ (Tk (̺δ ) − Tk (̺)) dx dt δ→0 0 Ω Z TZ = (2µ + λ) lim+ divx uδ (Tk (̺δ ) − Tk (̺)) + (Tk (̺) − Tk (̺)) dx dt δ→0 0 Ω h 2 ≤ C lim sup kdivx uδ kL ((0,T )×Ω) kTk (̺δ ) − Tk (̺)kL2 ((0,T )×Ω) δ→0+ i + kTk (̺) − Tk (̺)kL2 ((0,T )×Ω) . Moreover, kTk (̺) − Tk (̺δ )kL2 ((0,T )×Ω) kTk (̺) − Tk (̺)kL2 ((0,T )×Ω) ≤ lim inf + δ→0 ≤ lim sup kTk (̺) − Tk (̺δ )kL2 ((0,T )×Ω) . δ→0+ Hence lim sup kTk (̺δ ) − Tk (̺)kγ+1 Lγ+1 ((0,T )×Ω) δ→0+ Z TZ = lim sup |Tk (̺δ ) − Tk (̺)|γ+1 dx dt δ→0+ 0 Ω Z TZ ≤ ̺γ Tk (̺)−̺γ Tk (̺) dx dt 0 Ω ≤ C lim sup kdivx uδ kL2 ((0,T )×Ω) kTk (̺δ ) − Tk (̺)kL2 ((0,T )×Ω) δ→0+ ≤ C lim sup kdivx uδ kL2 ((0,T )×Ω) kTk (̺δ ) − Tk (̺)kLγ+1 ((0,T )×Ω) . δ→0+ As we control the L2 -norm of divx uδ , the proof of (i) is finished. We now prove (iii). Lemma 8.4 Let Q ⊂ R4 be an open set. Let ̺δ ⇀ ̺ in L1 (Q), uδ ⇀ u in Lr (Q; R3 ), ∇x uδ ⇀ ∇x u in Lr (Q; R3×3 ), (8.50) 104 CHAPTER 8. EXISTENCE PROOF where r > 1. Let oscq (̺δ − ̺) < ∞, (8.51) + 1r < 1, where ̺δ , uδ are renormalized solutions to the continuity equation. Then also the limit ̺, u is a renormalized solution to the continuity equation. 1 q Remark 8.2 The claim of the lemma considers the following definition of the renormalized solutions to the continuity equation: for any b ∈ C 1 ([0, ∞)) such that b′ (z) = 0 for z ≥ M for some M > 0 it holds ∂t (b(̺)) + divx (b(̺)u) + (b′ (̺)̺ − b(̺))divx u = 0 in D′ ((0, T ) × Ω). Proof: First of all, note that it is enough to show the result on J × K with J a bounded time interval, K a ball such that J × K ⊂ Q. Recall that we consider functions b(z) of class C 1 ([0, ∞)) which are constant for large values of z. Due to the assumption of the lemma and results from Chapter 7 we know that Tk (̺δ ) → Tk (̺) in Cweak (J; Lβ (K)) for any 1 ≤ β < ∞, Tk (̺δ )uδ ⇀ Tk (̺)u in Lr (J × K; R3 ). Therefore ∂t Tk (̺) + divx Tk (̺)u + ((Tk′ (̺)̺ − Tk (̺))divx u) = 0 in D′ (J × K). Proceeding as in the proof of Lemma 7.3 we can show that ′ ∂t b(Tk (̺)) + divx b(Tk (̺))u + (b (Tk (̺))Tk (̺) − b(Tk (̺)))divx u = −b′ (Tk (̺))((Tk′ (̺)̺ − Tk (̺))divx u) in D′ (J × K), where b′ (z) = 0 for z ≥ M . Note that lim lim k̺δ − Tk (̺δ )kL1 ((0,T )×Ω) = 0 k→∞ δ→0+ as ̺δ ⇀ ̺ in L1 ((0, T ) × Ω) and hence ̺δ is equiintegrable. On the other hand Z TZ Z TZ lim+ (̺δ − Tk (̺δ ))dx dt = (̺ − Tk (̺))dx dt δ→0 0 Ω 0 Ω = k̺ − Tk (̺)kL1 ((0,T )×Ω) . 8.5. ESTIMATES INDEPENDENT OF δ, LIMIT PASSAGE δ → 0+ 105 Therefore it suffices to show that b′ (Tk (̺))((Tk′ (̺)̺ − Tk (̺))divx u) → 0 in L1 (J × K) for k → ∞. Denote Qk,M = {(t, x) ∈ J × K; |Tk (̺)| ≤ M }. We have ′ b (Tk (̺))((Tk′ (̺)̺ − Tk (̺))divx u) L1 (J×K) ≤ C supδ>0 kdivx uδ kLr (J×K) lim inf δ→0 kTk (̺δ ) − Tk′ (̺δ )̺δ kLr′ (Qk,M ) . Clearly, kTk (̺δ ) − Tk′ (̺δ )̺δ kLr′ (Qk,M ) ≤ kTk (̺δ ) − Tk′ (̺δ )̺δ kαL1 (Qk,M ) kTk (̺δ ) − Tk′ (̺δ )̺δ kL1−α , q (Q k,M ) 0 < α < 1. As the family {̺δ }δ>0 is equiintegrable, due to a similar argument as above sup kTk (̺δ ) − Tk′ (̺δ )̺δ kL1 (J×K) → 0 δ>0 for k → ∞. Now, recalling that 0 ≤ Tk′ (̺δ )̺δ ≤ Tk (̺δ ), we get kTk (̺δ ) − Therefore Tk′ (̺δ )̺δ kLq (Qk,M ) ≤ kTk (̺δ ) − Tk (̺)kLq (Qk,M ) + kTk (̺) − Tk (̺)kLq (J×K) + kTk (̺)kLq (Qk,M ) 1 q q ≤ kTk (̺δ ) − Tk (̺)kL (Qk,M ) + oscq (̺δ − ̺) + M |J × K| . lim sup kTk (̺δ ) − Tk′ (̺δ )̺δ kLq (Qk,M ) δ→0 1 ≤ 2oscq (̺δ − ̺) + M |J × K| q ≤ C. The lemma is proved. 106 CHAPTER 8. EXISTENCE PROOF Next we take bk (̺) = ̺ note that b′k (̺) = Z ̺ 1 Z ̺ 1 Tk (z) dz; z2 Tk (̺) Tk (z) dz + , 2 z ̺ i.e. b′′k (̺) > 0 for ̺ > 0. Then ̺b′k (̺) − bk (̺) = Tk (̺) and we have (it follows by the limit passage δ → 0+ in the renormalized continuity equation for δ > 0) Z Z d bk (̺) dx + Tk (̺)divx u dx = 0 in D′ (0, T ) dt Ω Ω and, due to Lemma 8.4 Z Z d bk (̺) dx + Tk (̺)divx u dx = 0 in D′ (0, T ). dt Ω Ω Therefore Z Z tZ Tk (̺)divx u − Tk (̺)divx u dx dτ. bk (̺(t)) − bk (̺(t)) dx = Ω 0 Ω But bk is convex and thus Z TZ Tk (̺)divx u − Tk (̺)divx u dx dt 0≤ 0 Ω Z TZ = Tk (̺) − Tk (̺) divx u dx dt 0 Ω Z TZ + Tk (̺)divx u − Tk (̺)divx u dx dt. 0 Ω Now from (8.42) and (8.49) Z TZ Tk (̺)divx u − Tk (̺)divx u dx dt 0 Ω Z TZ 1 (̺γ Tk (̺) − ̺γ Tk (̺)) dx dt = 2µ + λ 0 Ω Z TZ 1 lim sup |Tk (̺δ ) − Tk (̺)|γ+1 dx dt, ≥ 2µ + λ δ→0+ 0 Ω 8.5. ESTIMATES INDEPENDENT OF δ, LIMIT PASSAGE δ → 0+ 107 i.e. Z T 1 lim sup kTk (̺δ ) − Tk (̺)kγ+1 Lγ+1 (Ω) 2µ + λ δ→0+ 0 Z TZ |Tk (̺) − Tk (̺)||divx u| dx dt ≤ 0 Ω ≤ kTk (̺) − Tk (̺)kL2 ((0,T )×Ω) kdivx ukL2 ((0,T )×Ω) γ−1 γ+1 ≤ CkTk (̺) − Tk (̺)kL2γ1 ((0,T )×Ω) kTk (̺) − Tk (̺)kL2γγ+1 ((0,T )×Ω) . Recall that kTk (̺) − Tk (̺)kL1 ((0,T )×Ω) ≤ kTk (̺) − ̺kL1 ((0,T )×Ω) + kTk (̺) − ̺kL1 ((0,T )×Ω) . Hence lim kTk (̺) − Tk (̺)kL1 ((0,T )×Ω) = 0. k→∞ As lim kTk (̺) − Tk (̺)kLγ+1 ((0,T )×Ω) ≤ oscγ+1 (̺δ − ̺) = C, k→∞ we also have that lim lim sup kTk (̺δ ) − Tk (̺)kLγ+1 ((0,T )×Ω) = 0. k→∞ δ→0+ Finally, as lim sup k̺δ − ̺kL1 ((0,T )×Ω) ≤ lim sup k̺δ − Tk (̺δ )kL1 ((0,T )×Ω) δ→0+ δ→0+ + lim sup kTk (̺δ ) − Tk (̺)kL1 ((0,T )×Ω) + lim sup kTk (̺) − ̺kL1 ((0,T )×Ω) = 0, δ→0+ δ→0+ we proved ̺δ → ̺ in L1 ((0, T ) × Ω) and therefore also in Lp ((0, T ) × Ω) for every p < γ + θ. To conclude the existence proof, note that we may pass to the limit in the energy inequality as before. We have Theorem 8.3 Let γ > 23 , 0 < Θ ≤ 1, Ω ∈ C 2,Θ , 0 < T < ∞ and ̺0 ∈ 2 0| ∈ L1 (Ω) and f ∈ L∞ ((0, T ) × Ω; R3 ). Let p(̺) = ̺γ . Lγ (Ω), ̺0 |u0 |2 = |(̺u) ̺0 Then there exists a weak solution to the compressible Navier–Stokes system satisfying the energy inequality, i.e. a weak solution in the sense of Chapter 5. 108 CHAPTER 8. EXISTENCE PROOF Bibliography [1] H. Amann. Linear and quasilinear parabolic problems, I. Birkhäuser Verlag, Basel, 1995. [2] G.-Q. Chen and H. Frid. On the theory of divergence-measure fields and its applications. Bol. Soc. Brasil. Mat. (N.S.), 32(3):401–433, 2001. Dedicated to Constantine Dafermos on his 60th birthday. [3] A. J. Chorin and J. E. Marsden. A mathematical introduction to fluid mechanics. Springer-Verlag, New York, 1979. [4] R.J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98:511–547, 1989. [5] D. B. Ebin. Viscous fluids in a domain with frictionless boundary. Global Analysis - Analysis on Manifolds, H. Kurke, J. Mecke, H. Triebel, R. Thiele Editors, Teubner-Texte zur Mathematik 57, Teubner, Leipzig, pages 93–110, 1983. [6] L. C. Evans. Partial differential equations. Graduate Studies in Math. 19, Amer. Math. Soc., Providence, 1998. [7] C. L. Fefferman. Existence and smoothness of the Navier-Stokes equation. In The millennium prize problems, pages 57–67. Clay Math. Inst., Cambridge, MA, 2006. [8] E. Feireisl. Dynamics of viscous compressible fluids. Oxford University Press, Oxford, 2004. [9] D. Hoff. Dynamics of singularity surfaces for compressible viscous flows in two space dimensions. Commun. Pure Appl. Math., 55:1365–1407, 2002. 109 110 BIBLIOGRAPHY [10] A. Kufner, O. John, and S. Fučı́k. Function spaces. Noordhoff International Publishing, Leyden, 1977. Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. [11] J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 63:193–248, 1934. [12] P.-L. Lions. Mathematical topics in fluid dynamics, Vol.2, Compressible models. Oxford Science Publication, Oxford, 1998. [13] F. Murat. Compacité par compensation. Ann. Sc. Norm. Sup. Pisa, Cl. Sci. Ser. 5, IV:489–507, 1978. [14] A. Novotný and I. Straškraba. Introduction to the mathematical theory of compressible flow. Oxford University Press, Oxford, 2004. [15] N. V. Priezjev and S.M. Troian. Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular versus continuum predictions. J. Fluid Mech., 554:25–46, 2006. [16] L. Tartar. Compensated compactness and applications to partial differential equations. Nonlinear Anal. and Mech., Heriot-Watt Sympos., L.J. Knopps editor, Research Notes in Math 39, Pitman, Boston, pages 136–211, 1975. [17] V. A. Vaigant and A. V. Kazhikhov. On the existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid (in Russian). Sibirskij Mat. Z., 36(6):1283–1316, 1995.

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement