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 9 2.1 Mass conservation......................... 9 2.2 Balance of momentum...................... 1 2.3 Spatial domain and boundary conditions............ 12 2.4 Initial conditions......................... 13 3 Weak solutions 15 3.1 Equation of continuity weak formulation........... 15 3.1.1 Weak-strong compatibility................ 16 3.1.2 Weak continuity...................... 16 3.1.3 Total mass conservation................. 17 3.2 Balance of momentum weak formulation........... 18 4 A priori bounds 21 4.1 Total mass conservation..................... 21 4.2 Energy balance.......................... 22 4.3 Pressure estimates........................ 24 5 Complete weak formulation 27 5.1 Equation of continuity...................... 27 5.2 Momentum equation....................... 28 5.3 Energy inequality......................... 29 3
4 CONTENTS II Weak sequential stability for large γ 31 6 Weak sequential stability 33 6.1 Uniform bounds.......................... 34 6.2 Limit passage........................... 35 6.3 Compactness of the convective term............... 36 6.4 Passing to the limit step 1.................. 36 6.5 Strong convergence of the densities............... 38 6.5.1 Compactness via Div-Curl lemma............ 38 6.5.2 Renormalized equation.................. 4 6.5.3 The effective viscous flux................. 42 III Existence of weak solutions for small γ 49 7 Mathematical tools 53 7.1 Continuity equation........................ 53 7.2 Continuity in time........................ 62 8 Existence proof 69 8.1 Approximations.......................... 69 8.2 Existence for the Galerkin approximation............ 7 8.3 Estimates independent of n, limit passage n....... 8 8.4 Estimates independent of ε, limit passage ε +....... 84 8.4.1 Strong convergence of the density............ 89 8.5 Estimates independent of δ, limit passage δ +....... 91 8.5.1 Strong convergence of the density............ 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 [12] constructed the weak solutions to the incompressible Navier Stokes system as early as in 193, 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 [13] who generalized Leray s theory to the case of barotropic compressible viscous fluids see also Vaigant and Kazhikhov [18]. 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 [13], to- 7
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...............................................ϱ = ϱt, x, the velocity field.............................................. u = ut, x, functions of the time t R and the spatial position x R 3. 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 R 3 containing a fluid of density ϱ. The change of the total mass of the fluid contained in B during a time interval [t 1, t 2 ], t 1 < t 2 is given as ϱt 2, x dx ϱt 1, x dx. 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 ϱu n, where 9 B
1 CHAPTER 2. MATHEMATICAL MODEL n denotes the outer normal vector to B: t2 ϱt 2, x dx ϱt 1, x dx = B B t 1 B ϱt, xut, x nx ds x dt. 2.1 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: t2 ϱt 2, x dx ϱt 1, x dx = div x ϱt, xut, x dx dt. B B t 1 B Furthermore, fixing t 1 = t and performing the limit t 2 t we may use the mean value theorem to obtain 1 t ϱt, x dx = lim ϱt 2, x dx ϱt, x dx 2.2 t2 t t 2 t B = lim t2 t 1 t 2 t = B t2 t B B div x ϱt, xut, x dx dt div x ϱt, xut, x dx. Finally, as relation 2.2 should hold for any volume element B, we may infer that t ϱt, x + div x ϱt, xut, x =. 2.3 Relation 2.3 is a first order partial differential equation called equation of continuity. B 2.2 Balance of momentum Using arguments similar to the preceding part, we derive balance of momentum in the form t ϱt, xut, x +div x ϱt, xut, x ut, x = div x Tt, x+ϱt, xft, x, 2.4
2.2. BALANCE OF MOMENTUM 11 or, equivalently cf. 2.3, [ ] ϱt, x t ut, x + ut, x x ut, x = div x Tt, x + ϱt, xft, 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 S = S x u = µ x u + t xu 2 3 div xui + ηdiv x ui, 2.5 with the shear viscosity coefficient µ and the bulk viscosity coefficient η, here assumed constant, µ >, η. 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 = pϱ. Accordingly, div x T = µ u + λ + µ x div x u x pϱ, µ >, λ 2 3 µ, and equations 2.3, 2.4 can be written in a concise form as Navier Stokes system t ϱ + div x ϱu =, 2.6 t ϱu + div x ϱu u + x pϱ = µ u + λ + µ x div x u + ϱf. 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 ϱ 1, while 2.6, 2.7 reduces to div x u =, 2.8
12 CHAPTER 2. MATHEMATICAL MODEL t u + div x 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 Navier Stokes 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 R 3. The presence of the physical boundary 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, ϱt, x = ϱt, x + a i, ut, x = ut, x + a i for all t, x, 2.1 where the period vectors a 1 = a 1,,, a 2 =, a 2,, a 3 =,, a 3 are given. Equivalently, we may assume that is a flat torus, = [, a 1 ] {,a1 } [, a 2 ] {,a2 } [, a 3 ] {,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 and satisfying the extra geometric restrictions: ϱt, x = ϱt, x, u i t,, x i, = u i t,, x i,, i = 1, 2, 3, and, similarly, u i t,, x j, = u i t,, x j, for i j, f i t,, x i, = f i t,, x i,, f i t,, x j, = f i t,, x j, for i j, we can check that the equations 2.6, 2.7 are invariant with respect to the above transformations;
2.4. INITIAL CONDITIONS 13 the velocity field u satisfies the so-called complete slip conditions u n =, [Sn] n = 2.11 on the boundary of the spatial block [, a 1 ] [, a 2 ] [, a 3 ]. We remark that the most commonly used boundary conditions for viscous fluids confined to a general spatial domain not necessarily a flat torus are the no-slip u =. 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 [16] for relevant discussion. 2.4 Initial conditions Given the initial state at a reference time t, say t =, 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 ϱ, x = ϱ x, x, 2.13 together with the initial distribution of the momentum, ϱu, x = ϱu x, x, 2.14 as, strictly speaking, the momentum balance 2.7 is an evolutionary equation for ϱu 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 ϱ may not be continuous. As shown by Hoff [1], 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, T, where is the flat torus introduced in Section 2.3. Multiplying 2.6 on φ Cc, T, integrating the resulting expression over, T, and performing by parts integration, we obtain T ϱt, x t φt, x + ϱt, xut, x x φt, x dx dt =. 3.1 15
16 CHAPTER 3. WEAK SOLUTIONS Definition 3.1 We say that a pair of functions ϱ, u is a weak solution to equation 2.6 in the space-time cylinder, T if ϱ, ϱu are locally integrable in, T and the integral identity 3.1 holds for any test function φ Cc, T. 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, Taking ϱu L 1, T ; L 1 ; R 3. 3.2 φt, x = ψtϕx, ψ C c, T, ϕ C c as a test function in 3.1 we may infer, by virtue of 3.2, that the function t ϱt, xϕx dx is absolutely continuous in [, T ] 3.3 for any ϕ Cc. In particular, the initial condition 2.13 may be satisfied in the sense that ϱt, xϕx dx = ϱ xϕx dx for any ϕ Cc. lim t + Now, take φ ε t, x = ψ ε tφt, x, φ C c [, T ],
3.1. EQUATION OF CONTINUITY WEAK FORMULATION 17 where ψ ε C c, τ, ψ ε 1, ψ ε 1 [,τ] as ε. Taking φ ε as a test function in 3.1 and letting ε, we conclude, making use of 3.3, that ϱτ, xφτ, x dx ϱ xφ, x dx 3.4 = τ ϱt, x t φt, x + ϱt, xut, x x φt, x dx dt for any τ [, T ] and any φ Cc [, T ]. 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 φ ε t, x = ϕ ε x, with ϕ ε C c B such that It is easy to see that ϱτ, xφ ε τ, x dx ϕ ε 1, ϕ ε 1 B as ε. ϱ xφ ε, x dx B ϱτ, x dx B ϱ x dx as ε, 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 τ τ ϱt, xut, x x ϕ ε x dx dt ϱt, xut, x n ds x dt. In other words, the weak solutions possess a normal trace on the boundary of the cylinder, 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 [, τ] in 3.4 we obtain ϱτ, x dx = ϱ x dx = M for any τ, 3.5 meaning, the total mass M of the fluid is a constant of motion. B
18 CHAPTER 3. WEAK SOLUTIONS 3.2 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 ϱ, u represent a weak solution to the momentum equation 2.7 in the set, T if the integral identity T ϱut, x t φt, x + ϱu ut, x : x φt, x 3.6 = +pϱt, xdiv x φt, x dx dt T µ x ut, x : x φt, x +λ + µdiv x ut, xdiv x φt, x ϱt, xft, x φt, x dx dt is satisfied for any test function φ C c, T ; R 3. Of course, we have tacitly assumed that all quantities appearing in 3.6 are at least locally integrable in, T. 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 L 2, T ; W 1,2 ; R 3. If R 3 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 =, 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 L 2, T ; W 1,2 ; R 3, where W 1,2 ; R 3 is the Sobolev space obtained as the closure of Cc ; R 3 in the W 1,2 -norm. On the other hand, for the periodic boundary conditions, we can allow test functions φ Cc, T ; R 3 and the velocity u L 2, T ; Wper; 1,2 R 3. Remark 3.1 We may get the weak-strong compatibility as in the case of continuity equation.
2 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 ϱ satisfies the equation of continuity that may be written in the form t ϱ + u x ϱ = ϱdiv x u. 4.1 This is a transport equation with the characteristic field defined Accordingly, 4.1 can be written as Consequently, we obtain d dt Xt, x = ut, X, X, x = x. d dt ϱt, Xt, = ϱt, Xt, div xut, Xt,. 21
22 CHAPTER 4. A PRIORI BOUNDS for any t [, T ]. inf ϱ, x exp t div x u L x,t ϱt, x sup ϱ, x exp t div x u L,T x 4.2 Unfortunately, the bounds established in 4.2 depend on div x u L which we have no information. Thus we may infer only that on ϱt, x, 4.3 provided ϱ, x in. Relation 4.3 combined with the total mass conservation 3.5 yields ϱt, L 1 = ϱ L 1, ϱ, = ϱ. 4.4 4.2 Energy balance Taking the scalar product of the momentum equation 2.4 with u we deduce the kinetic energy balance equation 1 1 t +div 2 ϱ u 2 x 2 ϱ u 2 u +div x pϱu pϱdiv x u div x Su+S : x u = ϱf u. 4.5 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 2.1 we get d dt 1 2 ϱ u 2 dx where, in accordance with 2.7, pϱdiv x u dx + S : x u dx = ϱf u dx, S : x u = µ x u 2 + λ + µ div x u 2 c x u 2, c >, 4.6
4.2. ENERGY BALANCE 23 provided λ + 2/3µ >. Seeing that ϱf u dx f ϱ ϱ u dx 1 2 f L,T ;R 3 ϱ dx + ϱ u 2 dx, we focus on the integral pϱdiv x u dx. Multiplying the equation of continuity 4.1 by b ϱ we obtain the renormalized equation of continuity t bϱ + div x bϱu + b ϱϱ bϱ div x u =. 4.7 Consequently, in particular, the choice ϱ pz bϱ = P ϱ ϱ 1 z 2 dz leads to Thus b ϱϱ bϱ = pϱ. pϱdiv x u dx = d P ϱ dx, dt and we deduce the total energy balance d 1 dt 2 ϱ u 2 + P ϱ We conclude with dx + S : x u dx = ϱf u dx. 4.8
24 CHAPTER 4. A PRIORI BOUNDS Energy estimates: sup ϱut, L 2 ;R 3 ce, T, f, 4.9 t [,T ] sup t [,T ] T P ϱt, dx ce, T, f, 4.1 ut, 2 W 1,2 ;R 3 dt ce, T, f, 4.11 where E denotes the initial energy 1 E = 2 ϱ u 2 + P ϱ dx. 4.3 Pressure estimates A seemingly direct way to pressure estimates is to compute the pressure in the momentum balance 2.7: pϱ = 1 div x t ϱu 1 div x div x ϱu u+ 1 div x div x S+ 1 div x ϱf, 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 [ B[ϱ] = B bϱ 1 ] bϱ dx and integrate by parts to obtain T = 1 T pϱ dx pϱbϱ dx dt 4.12 bϱ dx dt + T S : x B[ϱ] dx dt
4.3. PRESSURE ESTIMATES 25 T T ϱu u : x B[ϱ] dx ϱu t B[ϱ] dx dt ϱu B[ϱ]τ, ϱ u B[ϱ ] dx. Furthermore, we have t B[ϱ] = B [ div x bϱu + b ϱϱ bϱ div x u 4.13 1 ] b ϱϱ bϱ div x u dx. We recall the basic properties of the Bogovskii operator: Bogovskii operator: div x B[h] = h for any h L p, h dx =, 1 < p <, B[h] =, 4.14 B[h] W 1,p ;R 3 cp h L p, 1 < p <, 4.15 B[h] L q ;R 3 g L q ;R 3 4.16 for h L p, h = div x g, g n =, 1 < q <. As will be seen in the last part Chapter 8, we can show that for bϱ = ϱ θ the right-hand side is possible to estimate provided { γ θ min 2, 2 } 3 γ 1. 4.17 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 R 3 is either a bounded domain with Lipschitz boundary or a periodic box. For the sake of definiteness, we take the pressure in the form pϱ = aϱ γ, with a > 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 [,, b =, b r = whenever r M b. 5.2 We say that ϱ, u is a renormalized solution of the equation of continuity 2.3, supplemented with the initial condition, ϱ, = ϱ, if ϱ C weak [, T ]; L γ, ϱ, u L 2, T ; W 1,2 ; R 3, and the integral identity 27
28 CHAPTER 5. COMPLETE WEAK FORMULATION T ϱ + bϱ t φ+ϱ + bϱ u x φ+bϱ b ϱϱ div x uφ dx dt = ϱ + bϱ φ, dx 5.3 is satisfied for any φ Cc [, T and any b belonging to the class specified in 5.2. In particular, taking b we deduce the standard weak formulation of 2.3 in the form = τ for any τ [, T ] and any φ C c ϱτ, φτ, ϱ φ, dx 5.4 ϱ t φ + ϱu x φ dx dt [, T ]. In case of space-periodic boundary conditions we assume ϱ and u spaceperiodic, while for the homogeneous Dirichlet boundary conditions we assume u L 2, T ; W 1,2 R 3 ; R 3. Note that 5.4 actually holds on the whole physical space R 3 provided in case of the Dirichlet boundary conditions ϱ, u were extended to be zero outside. Note also that 5.4 implies that the initial condition ϱ, = ϱ is fulfilled. In case of the space periodic boundary conditions we may extend the functions outside due to the periodicity. 5.2 Momentum equation In addition to the previous assumptions we suppose that ϱu C weak [, T ]; L q ; R 3 for a certain q > 1, pϱ L 1, T. The weak formulation of the momentum equation reads:
5.3. ENERGY INEQUALITY 29 = τ τ ϱuτ, φτ, ϱu φ, dx 5.5 ϱu t φ + ϱu u : x φ + pϱdiv x φ µ x u : x φ + λ + µdiv x udiv x φ ϱf φ dx dt dx dt for any τ [, T ] and for any test function φ C c [, T ] ; R 3. Note that 5.5 already includes the satisfaction of the initial condition ϱu, = ϱu. 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: 1 τ 2 ϱ u 2 + P ϱ τ, dx + µ x u 2 + λ + µ div x u 2 dx dt 5.6 1 τ ϱu 2 + P ϱ dx + ϱf u dx dt 2ϱ for a.a. τ, T, where ϱ pz P ϱ = ϱ 1 z 2 dz.
3 CHAPTER 5. COMPLETE WEAK FORMULATION Some remarks are in order. To begin, given the specific choice of the pressure pϱ = aϱ γ and the fact that the total mass of the fluid is a constant of motion, the function P ϱ in 5.6 can be taken as P ϱ = a γ 1 ϱγ. Next, we need a kind of compatibility condition between ϱ and ϱu provided we allow the initial density to vanish on a nonempty set: ϱu = a.a. on the vacuum set {x ϱ x = }. 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 {ϱ ε, u ε } ε> of weak solutions of the compressible Navier Stokes system, emanating from the initial data we want to show that ϱ ε, = ϱ,ε, ϱu, = ϱu,ε, ϱ ε ϱ, u ε u as ε in a certain sense and at least for suitable subsequences, where ϱ, 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 1 ϱu,ε 2 + P ϱ,ε dx E, 6.1 2ϱ,ε where the constant E 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 and f. In accordance with the energy inequality 5.6, we get for any T < together with sup ϱ ε t, L γ c 6.2 t,t sup ϱ ε u ε t, L 2 ;R 3 c, 6.3 t,t T u ε t, 2 W 1,2 ;R 3 dt c, 6.4 where the symbol c stands for a generic constant independent of ε. Interpolating 6.2, 6.3, we get with ϱ ε u ε L q ;R 3 = ϱ ε ϱε u ε L q ;R 3 ϱ ε L 2γ ϱ ε u ε L 2 ;R 3, We conclude that q = 2γ γ + 1 > 1 provided γ > 1. sup t [,T ] ϱ ε u ε t, L q ;R 3, q = 2γ γ + 1. 6.5 Next, applying a similar treatment to the convective term in the momentum equation, we have ϱ ε u ε u ε L q ;R 3 3 = ϱ ε u ε L 2γ/γ+1 ;R 3 u ε L 6 ;R 3, with q = 6γ 4γ + 3.
6.2. LIMIT PASSAGE 35 Using the standard embedding relation W 1,2 L 6, 6.6 we may therefore conclude that T ϱ ε u ε u ε 2 L q ;R 3 3 dt c, q = 6γ 4γ + 3. 6.7 Note that 6γ 4γ + 3 > 1 as long as γ > 3 2. Finally, we have the pressure estimates see Chapter 8 for the proof: T for α = min{ γ 2, 2 3 γ 1}. T pϱ ε ϱ α ε dx dt = a ϱε γ+α dx dt c 6.8 6.2 Limit passage In view of the uniform bounds established in the previous section, we may assume that ϱ ε ϱ weakly-* in L, T ; L γ, 6.9 u ε u weakly in L 2, T ; W 1,2 ; R 3 6.1 passing to suitable subsequences as the case may be. Moreover, since ϱ ε satisfies the equation of continuity 5.4, relation 6.9 may be strengthened to see Chapter 7 ϱ ε ϱ in C weak [, T ]; L γ. 6.11 Let us recall that, in view of 6.9, relation 6.11 simply means { } { } t ϱ ε t, φ dx t ϱt, φ dx in C[, T ] for any φ C c.
36 CHAPTER 6. WEAK SEQUENTIAL STABILITY 6.3 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 and even ϱ ε u ε ϱu weakly-* in L, T ; L 2γ/γ+1 ; R 3 ϱ ε u ε ϱu in C weak [, T ] : L 2γ/γ+1 ; R 3, 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 ϱu = ϱu. This can be observed in several ways. Seeing that we deduce that W 1,2 L q compactly for 1 q < 6, L p W 1,2 compactly whenever p > 6 5. 6.13 In particular, relation 6.12 yields ϱ ε u ε ϱu in C[, T ]; W 1,2, which, combined with 6.1, gives rise to the desired conclusion For more details see again Chapter 7. ϱu = ϱu. 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 ϱ ε u ε u ε ϱu u weakly in L q, T ; R 3 3 for a certain q > 1. 6.14
6.4. PASSING TO THE LIMIT STEP 1 37 Summing up the previous discussion we deduce that the limit functions ϱ, u satisfy the equation of continuity = τ ϱτ, φτ, ϱ φ, dx 6.15 ϱ t φ + ϱu x φ dx dt for any τ [, T ] and any φ Cc [, T ], together with a relation for the momentum = τ ϱuτ, φτ, ϱu φ, dx 6.16 ϱu t φ + ϱu u : x φ + pϱdiv x φ dx dt τ µ x u : x φ + λ + µdiv x u div x φ dx dt for any test function φ C c [, T ] ; R 3. Here, we have also to assume at least weak convergence of the initial data, specifically, ϱ,ε ϱ weakly in L γ, 6.17 ϱu,ε ϱu weakly in L 1 ; R 3. Thus it remains to show the crucial relation pϱ = pϱ or, equivalently, ϱ ε ϱ a.a. in, T. 6.18 This will be carried over in a series of steps specified in the remaining part of this chapter.
38 CHAPTER 6. WEAK SEQUENTIAL STABILITY 6.5 Strong convergence of the densities In order to simplify presentation and to highlight the leading ideas, we assume that in particular γ 5, ϱ ε ϱ in C weak, T ; L γ, γ 5. 6.5.1 Compactness via Div-Curl lemma Div-Curl lemma, developed by Murat and Tartar [14], [17], represents an efficient tool for handling compactness in non-linear problems, where the classical Rellich Kondraschev argument is not applicable.
6.5. STRONG CONVERGENCE OF THE DENSITIES 39 Div-Curl lemma: Lemma 6.1 Let B R M be an open set. Suppose that v n v weakly in L p B; R M, as n, where w n w weakly in L q B; R M 1 p + 1 q = 1 r < 1. Let, moreover, for a certain s > 1. {div[v]} n=1 be precompact in W 1,s B, {curl[w]} n=1 be precompact in W 1,s B; R M M Then v n w n v w weakly in L r B. We give the proof only for a very special case that will be needed in the future, namely, we assume that div v n =, w n = x Φ n, Φ n dy =. 6.19 R M Moreover, given the local character of the weak convergence, it is enough to show the result for B = R M. By the same token, we may assume that all functions are compactly supported. We recall that a scalar sequence {g n } n=1 is precompact in W 1,s R M if g n = div[h n ], with {h n } n=1 precompact in L s R M ; R M. Now, it follows from the standard compactness arguments that Φ n Φ strongly in L q R M, x Φ = w.
4 CHAPTER 6. WEAK SEQUENTIAL STABILITY Taking φ Cc R M we have v n w n φ dy = v n x Φ n φ dy R M R M = v n x φφ n dy v x φφ dy R M R M = v wφ dy, R M which completes the proof under the simplifying hypothesis 6.19. 6.5.2 Renormalized equation We start with the renormalized equation 5.3 with bϱ = ϱ logϱ ϱ: T ϱ ε logϱ ε t ψ ϱ ε div x u ε ψ dx dt = ϱ,ε logϱ,ε dx 6.2 for any ψ Cc [, T, ψ = 1. Clearly, relation 6.2 is a direct consequence of 5.3. Repeating the procedure from Chapter 2 we can get ϱ ε logϱ ε t, dx + T ϱ ε div x u ε dx dt = ϱ,ε logϱ,ε dx. 6.21 Passing to the limit for ε in 6.21 and assuming ϱ,ε ϱ in L p for some p > 1 we get t ϱ log ϱt, dx + ϱdiv x u dx dτ = ϱ logϱ dx. 6.22 Our next goal is to show that the limit functions ϱ, u, besides 6.15, satisfy also its renormalized version. To this end, we use the procedure proposed by DiPerna and Lions [4], specifically, we regularize 6.15 by a family of regularizing kernels κ δ x to obtain: t ϱ δ + div x ϱ δ u = div x ϱ δ u [div x ϱu] δ, with v δ = κ δ v, where stands for spatial convolution.
6.5. STRONG CONVERGENCE OF THE DENSITIES 41 We easily deduce that t bϱ δ + div x bϱ δ u + b ϱ δ ϱ δ bϱ δ div x u = b ϱ δ div x ϱ δ u [div x ϱu] δ. Taking the limit δ and using Friedrich s lemma see Chapter 7; here we need that ϱ L 2, T and the procedure from Chapter 2 we get t ϱ logϱt, dx + ϱdiv x u dx dτ = ϱ logϱ dx; whence, in combination with 6.22, t ϱ logϱ ϱ logϱ t, dx+ ϱdiv x u ϱdiv x u dx dτ =. 6.23 Assume, for a moment, that we can show τ τ ϱdiv x u dx dt ϱdiv x u dx dt for any τ >, 6.24 which, together with lower semi-continuity of convex functionals, yields ϱ logϱ = ϱ logϱ. 6.25 In order to continue, we need the following standard result: Lemma 6.2 Suppose that ϱ ε ϱ weakly in L 3 Q, where ϱ logϱ = ϱ logϱ. Then ϱ ε ϱ in L 1 Q.
42 CHAPTER 6. WEAK SEQUENTIAL STABILITY Proof: Suppose that < δ ϱ. Consequently, because of convexity of z z logz, we have a.e. in Q M = {t, x Q; ϱt, x M, ϱ ε t, x M ε, 1} Therefore ϱ ε logϱ ε ϱ logϱ = logϱ + 1 ϱ ε ϱ + αϱ, ϱ ε ϱ ε ϱ 2 logϱ + 1 ϱ ε ϱ + 1 2M ϱ ε ϱ 2. ϱ ε ϱ 2 dx dt {δ ϱ} Q M 2M logϱ 1ϱ ε ϱ dx dt {δ ϱ} Q M +2M ϱ ε logϱ ε ϱ logϱ dx dt. {δ ϱ} Q M Since Q\Q M ϱ ε ϱ 2 dx dt for M uniformly in ε, 1, we conclude that and Now, since ϱ ε ϱ a.a. on the set {ϱ δ} for any δ >. ϱ ε ϱ a.a. on the set {ϱ = } { < ϱ < δ} as δ, we obtain the desired conclusion. In accordance with the previous discussion, the proof of strong pointwise convergence of {ϱ ε } ε> reduces to showing 6.24. This will be done in the next section. 6.5.3 The effective viscous flux The effective viscous flux 2µ + λdiv x u pϱ
6.5. STRONG CONVERGENCE OF THE DENSITIES 43 is a remarkable quantity that enjoys better regularity and compactness properties than its components separately. To see this, we start with the momentum equation ϱ ε u ε τ, φτ, ϱu,ε φ, dx 6.26 = τ τ ϱ ε u ε t φ + ϱ ε u ε u ε : x φ + pϱ ε div x φ dx dt µ x u ε : x φ + λ + µdiv x u ε div x φ dx dt, together with its weak limit ϱuτ, φτ, ϱu φ, dx 6.27 = τ τ ϱu t φ + ϱu u : x φ + pϱdiv x φ dx dt Our goal is to take µ x u : x φ + λ + µdiv x udiv x φ dx dt. φ = φ ε = ϕ x 1 [1 ϱ ε ], ϕ C c as a test function in 6.26, and φ = ϕ x 1 [1 ϱ], ϕ C c, in 6.27. Here, 1 represents the inverse of the Laplacean for space-periodic functions. Since R 3 is a bounded domain, we have x 1 [1 ϱ ε ] bounded in L, T ; W 1,γ ; R 3, γ > 3. Moreover, as 1 ϱ ε as well as 1 ϱ satisfy the equation of continuity, we have t x 1 [1 ϱ ε ] = x 1 div x [ϱ ε u ε ], t x 1 [1 ϱ] = x 1 div x [ϱu].
44 CHAPTER 6. WEAK SEQUENTIAL STABILITY Step 1: As ϱ ε ϱ in C weak [, T ]; L γ, we have, in accordance with the standard Sobolev embedding relation with W 1,γ C, x 1 [1 ϱ ε ] x 1 [1 ϱ] in C[, T ]. In particular, we deduce from 6.26, 6.27, τ ϱ ε u ε t φ ε + ϱ ε u ε u ε : x φ ε + pϱ ε div x φ ε dx dt lim ε τ = τ τ similarly for φ ε. Therefore lim ε τ τ lim ε = τ τ µ x u ε : x φ ε + λ + µdiv x u ε div x φ ε dx dt ϱu t φ + ϱu u : x φ + pϱdiv x φ dx dt µ x u : x φ + λ + µdiv x udiv x φ dx dt, φ = ϕ x 1 [1 ϱ], ϕpϱ ε ϱ ε + pϱ ε x ϕ x 1 [1 ϱ ε ] dx dt 6.28 ϕ µ x u ε : 2x 1 [1 ϱ ε ] + λ + µdiv x u ε ϱ ε dx dt τ lim ε µ x u ε ϕ x 1 [1 ϱ ε ] +λ + µdiv x u ε x ϕ x 1 [1 ϱ ε ] dx dt ϕpϱϱ + pϱ x ϕ x 1 [1 ϱ] dx dt ϕ µ x u : 2 x 1 [1 ϱ] + λ + µdiv x uϱ dx dt
6.5. STRONG CONVERGENCE OF THE DENSITIES 45 τ µ x u ϕ x 1 [1 ϱ] + λ + µdiv x u x ϕ x 1 [1 ϱ] dx dt τ + lim ϕϱ ε u ε x 1 [div x ϱ ε u ε ] ε ϱ ε u ε u ε : x ϕ x 1 [1 ϱ ε ] dx dt τ ϕϱu x 1 [div x ϱu] ϱu u : x ϕ x 1 [1 ϱ] dx dt. = Step 2: We have ϕ x u ε : 2 x 1 [1 ϱ ε ] dx = = i,j=1 ϕ 3 xj u i ε[ xi 1 xj ][1 ϱ ε ] dx i,j=1 3 xj ϕu i ε[ xi 1 xj ][1 ϱ ε ] dx i,j=1 ϕdiv x u ε ϱ ε dx+ x ϕ u ε ϱ ε dx lim ε τ 3 xj ϕu i ε[ xi 1 xj ][1 ϱ ε ] dx i,j=1 3 xj ϕu i ε[ xi 1 xj ][1 ϱ ε ] dx. Consequently, going back to 6.28 and dropping the compact terms, we obtain τ ϕ pϱ ε ϱ ε λ + 2µdiv x u ε ϱ ε dx dt 6.29 τ = lim ε ϕ pϱϱ λ + 2µdiv x uϱ dx dt ϕ ϱ ε u ε x 1 [div x ϱ ε u ε ] ϱ ε u ε u ε : x 1 x [1 ϱ ε ] dx dt τ ϕϱu x 1 [div x ϱu] ϱu u : x 1 x [1 ϱ] dx dt.
46 CHAPTER 6. WEAK SEQUENTIAL STABILITY Step 3: Our ultimate goal is to show that the right-hand side of 6.29 vanishes. To this end, we write ϱ ε u ε x 1 [div x ϱ ε u ε ] ϱ ε u ε u ε : x 1 x [1 ϱ ε ] = u ε ϱ ε x 1 [div x ϱ ε u ε ] ϱ ε u ε x 1 x [1 ϱ ε ]. Consider the bilinear form [v, w] = where we may write 3 i,j=1 v i R i,j [w j ] w i R i,j [v j ], R i,j = xi 1 xj, 3 i,j=1 v i R i,j [w j ] w i R i,j [v j ] where U i = = 3 i,j=1 v i R i,j [v j ]R i,j [w j ] w i R i,j [w j ]R i,j [v j ] 3 v i R i,j [v j ], W i = j=1 = U V W Z, 3 w i R i,j [w j ], div x U = div x W =, j=1 and 3 3 V i = xi 1 x jw j, Z i = xi 1 x jv j, i = 1, 2, 3. j=1 j=1 Thus a direct application of Div-Curl lemma Lemma 6.1 yields [v ε, w ε ] [v, w] weakly in L s R 3 whenever v ε v weakly in L p R 3 ; R 3, w ε w weakly in L q R 3 ; R 3, and 1 p + 1 q = 1 s < 1.
6.5. STRONG CONVERGENCE OF THE DENSITIES 47 Seeing that ϱ ε ϱ in C weak [, T ]; L γ, ϱ ε u ε ϱu in C weak [, T ]; L 2γ/γ+1 ; R 3 we conclude that 1 ϱ ε t, x 1 [div x ϱ ε u ε t, ] ϱ ε u ε t, x 1 x [1 ϱ ε t, ] 6.3 with ϱt, x 1 [div x ϱut, ] ϱut, x 1 x [1 ϱt, ] weakly in L s ; R 3 for all t [, T ], s = 2γ γ + 3 > 6 5 since γ 5. Thus we conclude that the convergence in 6.3 takes place in the space L q, T ; W 1,2 for any 1 q < ; whence, going back to 6.29, we conclude τ lim ε = τ ϕ pϱ ε ϱ ε λ + 2µdiv x u ε ϱ ε dx dt 6.31 ϕ pϱϱ λ + 2µdiv x uϱ dx dt. As a matter of fact, using exactly same method and localizing also in the space variable, we could prove that pϱϱ λ + 2µdiv x uϱ = pϱϱ λ + 2µdiv x uϱ, 6.32 which is the celebrated relation on weak continuity of the effective viscous pressure discovered by Lions [13]. Since p is a non-decreasing function, we have τ ϕ pϱ ε pϱ ϱ ε ϱ dx dt ;
48 CHAPTER 6. WEAK SEQUENTIAL STABILITY now relation 6.31 yields the desired conclusion 6.24, namely τ div x uϱ div x uϱ dx dt. Thus we get 6.25; whence ϱ ε ϱ a.a. in, T 6.33 and in L q, T for any q < γ + min { γ 2, 2 3 γ 1}.
Part III Existence of weak solutions for small γ 49
Last part of the lecture series is devoted to the proof of existence of weak solutions to the compressible Navier Stokes system provided pϱ ϱ γ with γ > 3. The proof is technically much more complicated than the previous 2 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 [15] by A. Novotný and I. Straškraba. For another approach, based on the construction of the approximative problem via a numerical method, see [9]. 51
52
Chapter 7 Mathematical tools 7.1 Continuity equation: renormalized solutions and extension We recall that for a function f L p R; L q R N, 1 p <, 1 q <, or f CR; L q R N we can define the mollifiers: over time we have T ε ft, x = ω ε t τfτ, x dτ; R T ε f C R; L q R N, T ε f f in L p R; L q R N if f L p R; L q R N, T ε f f in CR; L q R N if f CR; L q R N over space then S ε ft, x = ω ε x yft, y dy; R N S ε f L p R; C R N, S ε f f in L p R; L q R N if f L p R; L q R N 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 1,, 1 + 1 1. Let q β 1 α, 1 + 1 1. Assume for I R a bounded time interval p α Then ϱ L α I; L β loc RN, u L p I; W 1,q loc RN ; R N. S ε u x ϱ u x S ε ϱ in L s I; L r locr N. Here, 1 s = 1 α + 1 p and r [1, q] if β = and q 1, ], 1 q + 1 β 1 r 1 otherwise, where u x ϱ := div x ϱu ϱdiv x u in D R N. Proof: To simplify, we consider only the case β, q < which is enough for our purpose. Step 1: We have S ε u x ϱ, φ T = ϱt, yut, y x ω ε x y dy φt, x dx dt R N R N T R N ϱt, ydiv ut, yω ε x y dy R N φt, x dx dt, T u x S ε ϱ, φ = ut, x R N x ω ε x yϱt, y dy R N φt, x dx dt. Therefore T S ε u x ϱ u x S ε ϱ, φ = I ε t, x J ε t, x φt, x dx dt R N with I ε t, x = ϱt, y ut, y ut, x x ω ε x y dy, R N and J ε t, x = ϱt, ydiv x ut, yω ε x y dy. R N We define r as 1 = 1 r β + 1 q
7.1. CONTINUITY EQUATION 55 and get In Steps 2, 3 and 4 we show that J ε ϱdiv x u strongly in L s I; L r loc RN. I ε ϱdiv x u strongly in L s I; L r loc RN which will finish the proof of this lemma. Step 2: We aim at proving I ε L r BR C ϱt L β B R+1 x ut L q B R+2 ;R N N for a.a. t, T. We have I ε r L r B R = 1 x y ϱt, y ut, y ut, x ω dy dx B R x y ε εn+1 r ε = ut, x εz ut, x ϱt, x εz ωz dz dx r B R z 1 ε r ωz r r dz z 1 ϱt, x εz r ut, x εz ut, x dz dx r B R z 1 ε Cω ϱt, ξ r ut, ξ + εz ut, ξ dz dξ for ε < 1 r B R+1 z 1 ε q Cω B 1 r β ϱt, ξ B r ut, ξ + εz ut, ξ R+1 z 1 ε dz r q dξ r β Cω, N, q, β ϱt, ξ β dξ B R+1 q ut, ξ + εz ut, ξ r q ε dz dξ B R+1 z 1 C ϱt L β B R+1 x ut L q B R+2 ;R N N.
56 CHAPTER 7. MATHEMATICAL TOOLS Step 3: Let us show the strong convergence, first for a.a. t, T, in L r loc RN. Due to Step 2 it is enough to verify that the strong convergence holds for any ϱ C c R N, t, T fixed. Indeed, let ϱ n C c R N, ϱ n ϱt, in L β B R+1. Then I ε ϱdiv x ut, L r BR ϱt, y ϱ n y ut, y ut, x ω ε y dy R N + ϱ n y ut, y ut, x ω ε y dy ϱ n div x ut, R N + ϱ n ϱt, div x ut,. The first term is bounded by L r BR L r BR C ϱ n ϱt, L β B R+1 x ut, L q B R+2 ;R N N for n, the third is bounded by C ϱ n ϱt, L β B R div x ut, L q B R ;R N N for n. L r BR To conclude, let ϱ be a smooth function. Using the change of variables z = x y, as above, ε ut, x εz ut, x Ĩ ε t, x = ϱt, x εz ωz dz. ε z 1 As u W 1,q loc RN ; R N for a.a. t, T, ut, x εz ut, x ε = z 1 x ut, x ετz dτ z x ut, x for a.a. t, T and a.a. x, z R N B 1 a.a. points are Lebesgue points. Moreover, as ϱ is smooth, ϱt, x εz ϱt, x, x, z B R+1 B 1, t, T. Therefore, by Vitali s theorem Ĩεφt, x dx z i j ωz dz ϱt, x i u j t, xφt, x dx R N B 1 R N = ϱdiv x ut, xφt, x dx. R N
7.1. CONTINUITY EQUATION 57 Step 4: We have I ε s L s I;L r B R C T Due to this and the fact that ϱt, s L β B R+1 xut, s L q B R+2 ;R N N dt C ϱ s L α I;L β B R+1 xu s L p I;L q B R+2 ;R N N. I ε ϱdiv x u in L r loc RN for a.a. t, T, 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 R N. In particular, this shows that we may use as test functions smooth functions up to the boundary. Lemma 7.2 Let be a bounded Lipschitz domain in R N, N 2. ϱ L 2 Q T, u L 2 I; W 1,2 ; R N and f L 1 Q T satisfy Let t ϱ + div x ϱu = f in D Q T. Extending ϱ, u, f by,, outside, t ϱ + div x ϱu = f in D I R N. Proof: We have to show after the extension by,, Denote T = ϱ t η dx dt R N T T R N ϱu x η dx dt R N fη dx dt η C c, T R N. Φ m Cc, m N, Φ m 1, { Φ m x = 1 for x y ; disty, 1 }, m x Φ m x 2m for x.
58 CHAPTER 7. MATHEMATICAL TOOLS Evidently, Φ m 1 pointwise in and for any fixed compact K, We can write T T supp x Φ m \ K for m m K N, supp x Φ m. R N fη dx dt = R N ϱ t η dx dt = T We know that T T T T fηφ m dx dt + fη1 Φ m dx dt, R N R } N {{} T ϱ t Φ m η dx dt + ϱ t η1 Φ m dx dt, R N R } N {{} T ϱu x η dx dt = ϱu x Φ m η dx dt R N R N T T + ϱu x η1 Φ m dx dt ϱu x Φ m η dx dt. R } N R {{} N R N fηφ m dx dt = T R N ϱ t Φ m η dx dt T as Φ m η has support in. Therefore we have to show that T I m = ϱu x Φ m η dx dt. R N But due to the Hardy inequality T I m ϱ u x Φ m η dx dt R N T 2 sup ηt, x ϱ L u 2 {supp x Φ m } t,x distx, Cη, T ϱ L 2 {supp xφ m} u W 1,2 ;R N dt R N ϱu x Φ m η dx dt L 2 ;R N Cη, ϱ L 2,T ;L 2 {supp x Φ m } u L 2,T ;W 1,2 ;R N as m. The lemma is proved. dt
7.1. CONTINUITY EQUATION 59 Remark 7.1 Hence, in case ϱ L 2, T as u L 2, T ; W 1,2 ; R 3 will be satisfied, we further have ϱt, x dx = ϱs, x dx for any t, s [, T ]. It is enough to take η 1 in [s, t], provided ϱ is weakly continuous in L 1 which will be proved later. On the other hand, if only ϱ L p Q T, 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 =, 1 and ux = α, 1 x1 x ϱt, x = ux h x t 1 uy dy, 1 2 < α < 1, with h C 1 R, hs = for s. Evidently { u x α 1, x = α > 1 u W 1,2 2, 1,, u 1 x α 1, x 1 = α > 1, 2 ϱ C[, T ]; L p, 1, 1 p < 1 1 α u Lp, p < 1 α < 1, α and But t ϱ = 1 ux h t x x ϱu = h t x 1 dy uy 1 dy uy 1 ux ϱt, x dx is not constant, as ϱ, x dx =, but for h suitably chosen tϱ + x ϱu =. ϱt, x dx t >. 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
6 CHAPTER 7. MATHEMATICAL TOOLS Lemma 7.3 Let N 2, 2 β <, λ < 1, 1 < λ 1 β 2 1 and b C[, C 1,, b t ct λ, t [, 1], 7.1 b t ct λ 1, t 1. 7.2 Let ϱ L β I; L β loc RN, ϱ a.e. in I R N, u L 2 I; W 1,2 loc RN ; R N and f L z I; L z loc RN, z = if λ 1 >, z = 1 if λ 1. Suppose that β β λ 1 t ϱ + div x ϱu = f in D I R N. 7.3 i Then for any b C 1 [, satisfying 7.2 we have t bϱ + div x bϱu + {ϱb ϱ bϱ}div x u = fb ϱ in D I R N. 7.4 ii If f =, then 7.4 holds for any b satisfying 7.1 and 7.2. Proof: We consider only case ii, leaving i as possible exercise for the reader. We regularize 7.3 over space variable and get where But t S ε ϱ + div x S ε ϱu = r ε ϱ, u a.e. in I R N, 7.5 r ε ϱ, u = div x S ε ϱu div x S ε ϱu. r ε ϱ, u = u x S ε ϱ + S ε ϱdiv x u S ε div x ϱu = u x S ε ϱ S ε u x ϱ + S ε ϱdiv x u S ε ϱdiv x u, hence by Lemma 7.1 and an easy observation r ε ϱ, u in L r I; L r locr N, 1 r = 1 β + 1 1. 2 To avoid singularity at ϱ =, we multiply 7.5 by b h S εϱ with b h = bh +, h >, and obtain t b h S ε ϱ + div x b h S ε ϱu + S ε ϱb hs ε ϱ b h S ε ϱ div x u = r ε b hs ε ϱ a.e. in I R N.
7.1. CONTINUITY EQUATION 61 Now we pass with ε +. As S ε ϱ ϱ in L β I; L β loc RN i.e. for a subsequence a.e. in I R N, we get by Vitali s convergence theorem that b h S ε ϱ b h ϱ, S ε ϱb hs ε ϱ b h S ε ϱ ϱb hϱ b h ϱ in L p loc I RN, 1 p < 2 S ε ϱb h S εϱ CS ε ϱ 1+ β 2 1 for S ε ϱ 1. As this term is bounded also in L 2 I for bounded subset of R N, then the convergence holds also in L 2 weak I. Therefore, passing with ε, we have t b h ϱ + div x b h ϱu + ϱb hϱ b h ϱ div x u =, as T r ε b hs ε ϱ dx dt T r ε L r b hs ε ϱ L r dt, where 1 = 1 1 = 1 1 = β 2 and r r 2 β 2β b h S εϱ L r S ε ϱ L β. Finally we aim to pass with h +. Recall that {t, x; ϱ k} I k β ϱ β L β I {ϱ k}. Then we write for ψ C c I R N I R N = ϱb hϱ b h ϱ div x uψ dx dt ϱb hϱ b h ϱ div x uψ dx dt I R N {ϱ k} supp ψ + ϱb hϱ b h ϱ div x uψ dx dt. I R N {ϱ>k} supp ψ Now, passing with h +, the first term on the right-hand side goes to I R N ϱb ϱ bϱ div x uψ1 {ϱ k} dx dt,
62 CHAPTER 7. MATHEMATICAL TOOLS due to the Lebesgue dominated convergence theorem. The second term can be controlled by C ϱϱ + h β 2 1 + ϱ β 2 div x u ψ dx dt {ϱ>k} C ϱ β 2 + ϱ div x u ψ dx dt Further, I R N {ϱ k} C ϱ β 2 div L β I {ϱ k} xu L 2 I + ϱ β 2 L β I {ϱ k} k1 β 2 divx u L 2 I ϱb ϱ bϱ div x uψ1 {ϱ k} dx dt k I R N k. ϱb ϱ bϱ div x 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 C weak [, T ]; L q, 1 q <, if gφ dx C[, T ] for all φ Lq. We have the following very easy result: Lemma 7.4 Let 1 < q <, R N, I R be bounded. Let f L I; L q and t fη dx L1 I for all η Cc. Then there exists g C weak I; L q such that for a.a. t I ft, = gt, in the sense of L q. Proof: Take any η Cc. As fη dx W 1,1 I, after a possible change on times of measure zero we have that fη dx CI. Then it is easy to see that sup t I ft, L q ess sup ft, L q. t I
7.2. CONTINUITY IN TIME 63 Choose ε > and take arbitrary φ L q. Since Cc is dense in L q, 1 q <, we have ft + δ, ft, φ dx ft + δ, ft, η dx + ft + δ, ft, φ η dx, where η Cc is suitably chosen in such a way that the second integral is less than ε/2. Now, due to the continuity of fη dx, we can choose δ sufficiently small that for δ δ 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 ϱ L, T ; L γ and u L 2, T ; W 1,2 ; R 3, we immediately see at least for γ > 6 that ϱ C 5 weak[, T ]; L γ, as t ϱη dx = ϱu x η dx L 1 I, I [, T. Remark 7.3 Similarly we have that ϱu C weak [, T ]; L 2γ γ+1 ; R 3. Indeed, ϱ u 2γ γ+1 dx = ϱ u 2 γ γ+1 ϱ γ γ+1 dx ϱ u 2 dx γ γ+1 1 ϱ γ γ+1 dx L I. Looking at the weak formulation of the momentum equation, it is an easy task to verify that for φ C c [, T ] ; R 3 which finishes the proof. t ϱu φ dx L 1 I, In what follows we will use the following abstract version of the Arzelà Ascoli theorem see [11, Theorem 1.6.9]
64 CHAPTER 7. MATHEMATICAL TOOLS Theorem 7.1 Let X and B be Banach spaces such that B X. Let f n be a sequence of functions: I B which is uniformly bounded in B and uniformly continuous in X. Then there exists f CI; B such that f n f in CI; X at least for a chosen subsequence. Then we have Theorem 7.2 Let 1 < p, q <, be a bounded Lipschitz domain in R N, N 2. Let {g n } n=1 be a sequence of functions: I L q such that g n C weak I; L q g n is uniformly continuous in W 1,p g n is uniformly bounded in L q. Then at least for a chosen subsequence i g n g in C weak I; L q. ii If moreover L q W 1,p i.e. 1 < p N < p <, Np < q <, then N 1 N+p g n g in CI; W 1,p. Proof: i As W 1,p W 1,s for s = min N N 1 { p, N N 1 and 1 < q <, or }, the sequence g n is uniformly continuous in W 1,s. As the embedding L q W 1,s is compact, we have by virtue of Theorem 7.1 g n g in CI; W 1,s, at least for a chosen subsequence. Therefore, for a given ε > there exists n such that for m, n > n : g n t, g m t, η dx g n t, g m t, W 1,s η W 1,s ε η W 1,s, for all η Cc, for all t I. Hence for any η Cc the mappings t g nt, η dx form a Cauchy sequence in CI which has a limit A η CI. As sup A η t lim sup g n t, η dx C η L q, t I n
7.2. CONTINUITY IN TIME 65 η Cc, we see that η A η is a linear densely defined bounded operator from L q to R. Hence A η t = gt, η dx with gt, L q. Moreover, t gt, η dx CI for all η C c and by the density argument also for η L q. Moreover, again by the density argument g n t, gt, η dx n, hence sup t I g n t, η dx gt, η dx in CI for any η L q. To prove ii, recall that L q W 1,p and the result follows from Theorem 7.1. Next Lemma 7.5 Let be a bounded Lipschitz domain in R N, N 2, 1 < q <, 1 p <. If g n g in C weak I; L q, then g n g strongly in L p I; W 1,r provided L q W 1,r. Proof: As and L q W 1,r, we have g n t, gt, in L q, t I, g n t, gt, in W 1,r, t I. As in particular g n is bounded in L I; L q, then also sup g n t, L q C t I and so is sup t I g n t, W 1,r. Thus by the Lebesgue dominated convergence theorem T We further have g n t, gt, p W 1,r dt n.