SOME ANALYTICAL ISSUES FOR THE SELECTED COMPLEX FLUIDS MODELS

Transcription

1 SOME ANALYTICAL ISSUES FOR THE SELECTED COMPLEX FLUIDS MODELS by Cheng Yu B.S., Donghua University, Shanghai, China, June 22 M.S., Donghua University, Shanghai, China, March 25 M.A., Indiana University-Bloomington, December 29 Submitted to the Graduate Faculty of the Kenneth P. Dietrich Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 213

2 UNIVERSITY OF PITTSBURGH KENNETH P. DIETRICH ARTS AND SCIENCES This dissertation was presented by Cheng Yu It was defended on April 1, 213 and approved by Prof. Dehua Wang, Department of Mathematics. Prof. Giovanni P. Galdi, Department of Mechanical Engineering and Materials Science. Prof. Marta Lewicka, Department of Mathematics. Prof. Huijiang Jiang, Department of Mathematics. Prof. Noel J. Walkington, Department of Mathematics,Carnegie Mellon University. Dissertation Director: Prof. Dehua Wang, Department of Mathematics. ii

3 Copyright c by Cheng Yu 213 iii

4 ACKNOWLEDGMENTS After three years hard working in Pittsburgh, I am going to finish my graduate study. It would not have been possible without the kind support and help of many individuals and organizations. It is a good time to express my deep graduates to all of them who made my graduate study and dissertation successful. First and foremost, I am truly and deeply indebted to my advisor, Prof. Dehua Wang, for his many invaluable guidance, discussions and suggestions, and constant support and encouragements throughout my Ph. D study. Without his help, I cannot complete this dissertation and finish the Ph. D study. It is very good for me to work with him in my early career. I deeply appreciate him. I am grateful to my committee members Prof. Giovanni P. Galdi, Prof. Marta Lewicka, Prof. Huiqiang Jiang and Prof. Noel Walkington for their valuable time and effort for serving in the dissertation committee with many valuable comments. My sincere thanks go to Prof. Gui-Qiang Chen for providing me his insight on mathematics and his strong support. I would like to acknowledge my friends Shiting Bao, Ming Chen, Jilong Hu, Xianpeng Hu, Guoqing Liu, Qing Liu, Zhen Qin, Lizhi Ruan,Antonio Suen, Muhu Wang, Yuqi Wu,Sheng Xiong, Xiang Xu, Song Yao, Matthias Youngs, Rongfang Zhang, and many many other friends in Pittsburgh, in Bloomington, in China, in US, for their helps and encouragements during my early career. Thanks go to all my teachers, in particular, Prof. Guozhu Gao in mathematics, and Prof. Quanming Li in philosophy. I owe a great deal to the Mathematics Department and the University of Pittsburgh for providing me such a great place to learn, do and enjoy Mathematics. I cannot close these acknowledgements without thanking to my parents for their constant encouragement, strong support and believe in me. Special thanks are to my wife, Qing Xu, for her encouragement, support, love, and much much more. Thanks Mia to be our daughter and bring us many many joy time. iv

5 SOME ANALYTICAL ISSUES FOR THE SELECTED COMPLEX FLUIDS MODELS Cheng Yu, PhD University of Pittsburgh, 213 In this dissertation, we study the selected models from complex fluids: compressible flow of liquid crystals and the incompressible fluid-particles flow. On the compressible flow of liquid crystals, we establish the global existence of renormalized weak solutions when γ > 3 through a 2 three-level approximation, energy estimates, and weak convergence methods in the spirit of the so-called Lions-Feireisl method. On the incompressible fluid-particles flow, we establish the global existence of Leray weak solutions which was constructed by the Galerkin methods, fixed point arguments, and convergence analysis with the large initial data. The uniqueness was established by the classical theory of Stokes equations and a bootstrap argument in the two dimensional space. Keywords: Global weak solutions, existence, uniqueness, liquid crystal, Navier-Stokes equations, Vlasov equation. v

6 TABLE OF CONTENTS 1. INTRODUCTION Compressible flow of liquid crystals Incompressible fluid-particles flow Density-dependent incompressible fluid-particle flow COMPRESSIBLE FLOW OF LIQUID CRYSTALS Energy Estimates and Main Results The Solvability of the Direction Vector The Faedo-Galerkin Approximation Scheme Vanishing Viscosity Limit Uniform estimates of the density The vanishing viscosity limit passage The strong convergence of the density Passing to the Limit in the Artificial Pressure Term Better estimate of density The limit passage The strong convergence of density INCOMPRESSIBLE FLUID-PARTICLE FLOWS A Priori Estimates and Main Results The Existence of Weak Solutions Approximation Scheme Passing to the Limit as m Passing the limit as λ vi

7 3.3 Uniqueness in The Two Dimensional Space Regularity Uniqueness of solutions DENSITY-DEPENDENT FLUID-PARTICLE FLOW A Priori Estimates and Main Results Existence of Global Weak Solutions Construction of approximation solutions Pass to the limit as ε Pass the limit as δ BIBLIOGRAPHY vii

8 1. INTRODUCTION In this dissertation, our interests include the rigorous mathematical study of fluid models derived from the Navier-Stokes theory. In particular, we have been working on the compressible flow of nematic liquid crystals and the incompressible fluid-particles flow. Understanding these is a fundamental challenge in both mathematics and science. Our principal goal is to develop new analytic methods to tackle the mathematical issues and to gain new physical insights into the above flows and related applications. More precisely, the focus of this dissertation includes: Renormalized weak solutions to the compressible flow of nematic liquid crystals. Leray weak solutions to the incompressible fluid-particles flow, including global existence in three dimensions, and uniqueness in two dimensions. 1.1 COMPRESSIBLE FLOW OF LIQUID CRYSTALS The various applications of liquid crystals motivate us to investigate the related mathematical problems. The motion of nematic liquid crystals is governed by the forced Navier-Stokes equations and a parabolic type equation. The hydrodynamic equations for the three-dimensional flow of nematic liquid crystals ([1, 27, 34])has the following form: ρ t + div(ρu) =, (1.1.1a) (ρu) t + div(ρu u) + P (ρ) = µ u λdiv ( d d ( 12 ) d 2 + F (d))i 3, (1.1.1b) d t + u d = θ( d f(d)). (1.1.1c) 1

9 The system (1.1.1) is subject to the following initial-boundary conditions: (ρ, ρu, d) t= = (ρ (x), m (x), d (x)), x, (1.1.2) and u =, d = d (x), (1.1.3) where ρ L γ (), ρ ; d L () H 1 (); m L 1 (), m = if ρ = ; m 2 ρ L 1 (). Here R 3 is a smooth boundary domain, ρ is the density of fluid, u R 3 is the velocity of fluid, d R 3 is the direction field for the averaged macroscopic molecular orientations, and P = aρ γ is the pressure with constants a > and γ 1. The constants µ >, λ >, θ > denote the viscosity, the competition between kinetic energy and potential energy, and the microscopic elastic relation time for the molecular orientation field, respectively. The notation denotes the Kronecker tensor product, I 3 is the 3 3 identity matrix, and d d denotes the 3 3 matrix whose ij-th entry is < xi d, xj d >. The penalty function f(d) is the vector-valued smooth function and has the following form: f(d) = d F (d), where the scalar function F (d) denotes the bulk part of the elastic energy. Typically, we choose F (d) as the Ginzburg-Landau penalization thus yielding the penalty function f(d) as: F (d) = 1 ( d 2 1) 2, f(d) = 1 ( d 2 1)d, 4σ 2 2σ 2 where σ > is a constant. We refer the readers to [7, 1, 16, 27, 33, 34] for more mathematical models and physical background of liquid crystals. The first objective of this dissertation is to establish the existence of global weak solutions to (1.1.1)-(1.1.3). There have been extensive mathematical results on the incompressible flows of liquid crystals, for example, the existence of global weak solutions with large data, the global existence of strong solutions, and the partial regularity of the weak solutions similar to the 2

10 classical theorem of Caffarelli-Kohn-Nirenberg [5], see [3, 34, 35, 36, 37, 51] and the references cited therein. The existence of weak solutions to the density-dependent incompressible flow of liquid crystals was proved in [32]. The three dimensional compressible flow (1.1.1)-(1.1.3) of liquid crystals is much more complicated and difficult to establish the global existence due to strong nonlinearity. In the one-dimensional case the global existence of smooth and weak solutions to the compressible flow of liquid crystals was obtained in [11]. When the direction field d is absent in the system (1.1.1), the system reduces to the compressible Navier-Stokes equations. For the multidimensional compressible Navier-Stokes equations, Lions in [39] proved the global existence of finite energy weak solutions for γ > 9/5 by pioneering the concept of renormalized solutions to overcome the difficulties of large oscillations, and then Feireisl, et al, in [21, 18, 19] developed this method and extended the existence results to γ > 3/2. We shall study the initial-boundary value problem (1.1.1)-(1.1.3) for liquid crystals with large initial data in certain functional spaces with γ > 3/2. To achieve our goal, We shall employ a three-level approximation scheme similar to that in [21, 18] to prove the global existence, which consists of Faedo-Galerkin approximation, artificial viscosity, and artificial pressure. Then, in sprite of the work of [18], we prove that the uniform estimate of the density ρ γ+α in L 1 for some α > guarantees the vanishing of artificial pressure and the strong compactness of the density. We adopt the methods of Lions and Feireisl in [21, 18, 36] to build the weak continuity of the effective viscous flux for the compressible flow of liquid crystals similar to that for compressible Navier-Stokes equations to remove the difficulty of possible large oscillation of the density. To obtain the related estimates on effective viscous flux, we need to establish some estimates to deal with the direction field and its coupling and interaction with the fluid variables. 1.2 INCOMPRESSIBLE FLUID-PARTICLES FLOW On physical grounds, the motivation of our study of the incompressible fluid-particle flow is of primary importance in the modeling of sprays. There are many relevant applications, such as combustion theory, pollutant transport, and many more. The flow of the continuous phase is modeled by the forced Navier-Stokes equations, and the flow of the particles is governed by the 3

11 kinetic equation. The fluid-particle interactions are described by a friction force exerted from the fluid onto the particles. The second objective of this dissertation is to establish the global existence of weak solutions for the following partial differential equations, namely Navier-Stokes-Vlasov equations: u t + (u )u + p µ u = (u v)f dv, R d divu =, (1.2.1) f t + v x f + div v ((u v)f) =, in R d (, T ), where R d is a bounded domain, d = 2, 3, u is the velocity of the fluid, and p is the pressure. Without loss of generality, we take kinematic viscosity of fluid µ = 1 throughout the paper. The distribution function f(t, x, v) depends on the time t [, T ], the physical position x, and the velocity of particle v R d. The notation f(t, x, v) dv is the number of particles enclosed at t and location x in the volume element dv. The system is completed by the initial data u(, x) = u (x), f(, x, v) = f (x, v), (1.2.2) and with the following boundary conditions: u = on, and f(t, x, v) = f(t, x, v ) for x, v ν(x) < (1.2.3) where v = v 2(v ν(x))ν(x) is the specular velocity, ν(x) is the outward normal to. In general, the mathematical analysis of fluid-particle flow is challenging because the distribution function f depends on more variables than the fluid density ρ and velocity u. The rigorous mathematical study to such coupled systems is far from being complete but recently has received much attention. The global existence of weak solutions to Stokes-Vlasov system with boundary was established in 199s, see [26]. The existence theorem for weak solutions has been extended to Navier-Stokes-Vlasov equations within a periodic domain in [4]. The global existence of smooth solutions for Navier-Stokes-Vlasov-Fokker-Planck equations with small data was proved in [23]. More Recently, the existence of global weak solutions of Navier-Stokes-Vlasov-Poisson system with corresponding boundary value problem was established in [1]. Meanwhile, there are many 4

12 works in the direction of hydrodynamic limits, we refer the reader to [6, 24, 25, 46]. In works [6, 24, 25, 46], the authors used some scaling issues and convergence methods to investigate the hydrodynamic limits. A key idea in [24, 25] is to control the dissipation rate of a certain free energy associated with the whole space. For the compressible version, local strong solutions of Euler-Vlasov equations was established in [2]. Global existence of weak solutions for compressible Navier-Stokes equations coupled to Vlasov-Fokker-Planck equations was established in [45]. In Section 3, we shall establish the global existence of weak solutions to the initial-boundary value problem (1.2.1)-(1.2.3) for Navier-Stokes-Vlasov equations with large data in three dimensional space. To this end, we construct a new approximation scheme motivated by the works of [13, 44, 45]. The key idea of this approximation is to control the modified force term of regularized Navier-Stokes equations. The existence and uniqueness of the modified Vlasov equation is classically obtained, for example, see [3, 12, 26]. The controls of fdv and vf dv ensure R d R d that the modified Navier-Stokes equations could be solved. The compactness properties of the system will allow us to pass the limit to recover the original system. We shall also establish the uniqueness of the weak solutions in the two dimensional space. 1.3 DENSITY-DEPENDENT INCOMPRESSIBLE FLUID-PARTICLE FLOW Now, let us move to the Navier-Stokes-Vlasov equations for particles dispersed in a densitydependent incompressible viscous fluid: ρ t + div(ρu) =, (1.3.1) (ρu) t + div(ρu u) + p µ u = m p F fdv, R 3 (1.3.2) divu =, (1.3.3) f t + v x f + div v (F f) =, (1.3.4) 5

13 for (x, v, t) in R 3 (, ), where R 3 is a bounded domain, ρ is the density of the fluid, u is the velocity of the fluid, p is the pressure, µ is kinematic viscosity of fluid. The density distribution function f(t, x, v) of particles depends on the time t [, T ], the physical position x and the velocity of particle v R 3. In (1.3.2), m p is the mass of the particle and F is the drag force. The interaction of the fluid and particles is through the drag force exerted by the fluid onto the particles. Typically, the drag force F depends on the relative velocity u v and on the density of fluid ρ (e.g. [46]), such as F = F ρ(u v), (1.3.5) where F is a positive constant. Without loss of generality we take µ = F = m p = 1 throughout the paper. The final objective is to establish the global existence of weak solutions to the initial-boundary value problem for the system (1.3.1)-(1.3.5) subject to the following initial data: ρ t= = ρ (x), x, (1.3.6) (ρu) t= = m (x), x, (1.3.7) and the following boundary conditions: f t= = f (x, v), x, v R N, (1.3.8) u(t, x) = on, f(t, x, v) = f(t, x, v ) for x, v ν(x) <, (1.3.9) where v = v 2(v ν(x))ν(x) is the specular velocity, and ν(x) is the outward normal vector to. When the drag force is assumed independent on density in (1.3.5), hydrodynamic limits and the global existence of weak solutions to the Navier-Stokes and Vlasov-Fokker-Planck equations were studied in [24, 25, 45, 46]. When the drag force depends on the density as in (1.3.5), a relaxation of the kinetic regime toward a hydrodynamic regime with velocity u on the vacuum {ρ = } can not be excepted. It is difficult to establish a priori lower estimates on the density from the mathematics view point. 6

14 2. COMPRESSIBLE FLOW OF LIQUID CRYSTALS The global existence of weak solutions with large initial data to the compressible flows is always a basic and interesting problem of the mathematical study. The goal of this chapter is to study the global existence of weak solutions to the three dimensional compressible flow of liquid crystals with bounded domain. The remaining part of this chapter is organized as follows. In Section 2.1, after deduce the basic energy law, we state the main existence result of this chapter. In the following Sections, we use the three-level approximations, namely Faedo-Galerkin, vanishing viscosity, and artificial pressure, respectively, to prove our main result. 2.1 ENERGY ESTIMATES AND MAIN RESULTS In this section, we derive some basic energy estimates for the initial-boundary problem (1.1.1)- (1.1.3), introduce the notion of finite energy weak solutions in the spirit of Feireisl [21, 18], and state the main results. Without loss of generality, we take θ = a = 1. First we formally derive the energy equality and some a priori estimates, which will play a very important role in our paper. Multiplying (1.1.1b) by u, integrating over, and using the boundary condition (1.1.3), we obtain ( ) 1 t 2 ρ u 2 + ργ γ 1 = λ div dx + µ u 2 dx ( d d ( 12 ) d 2 + F (d))i 3 udx. 7

15 Using the equality we have Hence, we obtain div( d d) = ( 1 2 d 2 ) + ( d) d, = div ( d d ( 12 ) d 2 + F (d))i 3 udx ( d) d udx d F (d)udx. ( ) 1 t 2 ρ u 2 + ργ dx + u 2 dx γ 1 = λ ( d) d udx + λ d F (d)udx. (2.1.1) Multiplying by λ( d f(d)) on the both sides of (1.1.1c) and integrating over, we get t = λ λ d 2 dx t λf (d)dx λ d F (d)udx + λ ( d) d udx 2 d f(d) 2 dx. Then, from (2.1.1), we have the following energy equality to the system (1.1.1), ( 1 t 2 ρ u 2 + ργ γ 1 + λ ) 2 d 2 + λf (d) dx ( + µ u 2 dx + λ d f(d) 2) dx =. (2.1.2) Set E(t) = ( 1 2 ρ u 2 + ργ γ 1 + λ ) 2 d 2 + λf (d) (t, x)dx, and assume that E() <. From (2.1.2), we have the following a priori estimates: ρ u 2 L ([, T ]; L 1 ()); ρ L ([, T ]; L γ ()); d L ([, T ]; L 2 ()); F (d) L ([, T ]; L 1 ()); 8

16 u L 2 ([, T ]; L 2 ()); and also d f(d) L 2 ([, T ]; L 2 ()). (2.1.3) Although the above estimates will play very important roles in proving of our main existence theorem, they cannot provide sufficient regularity for the direction field d to control the strongly nonlinear terms containing d. Remark We infer from Gagliardo-Nirenberg inequality that d L 1 C d 4 5 L 6 d 1 5 L 2 + C d L 6 C d 4 5 H 1 d 1 5 L 2 + C d H 1, d L 1 3 C d 2 5 L 2 d 3 5 L 2 + C d L 2. Using d L (, T ; H 1 ()) and d L 2 (, T ; L 2 ()), we will have d L 1 (, T ; ) and d L 1 3 (, T ; ). Through our paper, we will use C to denote a generic positive constant, D to denote C, and D to denote the sense of distributions. To introduce the finite energy weak solution (ρ, u, d), we also need to take a differentiable function b, and multiply (1.1.1a) by b (ρ) to get the renormalized form: b(ρ) t + div(b(ρ)u) + (b (ρ)ρ b(ρ))divu =. (2.1.4) We define the finite energy weak solution (ρ, u, d) to the initial-boundary value problem (1.1.1)- (1.1.3) in the following sense: for any T >, ρ, ρ L ([, T ]; L γ ()), u L 2 ([, T ]; W 1,2 ()), d L ((, T ) ) L ([, T ]; H 1 ()) L 2 ([, T ]; H 2 ()), with (ρ, ρu, d)(, x) = (ρ (x), m (x), d (x)) for x ; The equations (1.1.1) hold in D ((, T ) ), and (1.1.1a) holds in D ((, T ) R 3 ) provided ρ, u are prolonged to be zero on R 3 \ ; 9

17 (2.1.4) holds in D ((, T ) ), for any b C 1 (R + ) such that b (z) = for all z R + large enough, say z M, (2.1.5) where the constant M may vary for different function b; The energy inequality E(t) + t holds for almost every t [, T ]. ( µ u 2 dx + λ d f(d) 2) dxds E() Remark It s possible to deduce that (2.1.4) will hold for any b C 1 (, ) C[, ) satisfying the following conditions b (z) c(z α + z γ 2 ) for all z > and a certain α (, γ 2 ) (2.1.6) provided (ρ, u, d) is a finite energy weak solution in the sense of the above definition (see details in [18]). Now, our main result on the existence of finite energy weak solutions reads as follows: Theorem Assume that R 3 is a bounded domain of the class C 2+ν, ν >, and γ > 3 2. then for any given T > the initial-boundary value problem (1.1.1)-(1.1.3) has a finite weak energy solution (ρ, u, d) on (, T ). The proof of Theorem is based on the following approximation scheme: ρ t + div(ρu) = ε ρ, (2.1.7a) (ρu) t + div(ρu u) + P (ρ) + δ ρ β + ε u ρ = µ u λdiv ( d d ( 12 ) d 2 + F (d))i 3, (2.1.7b) d t + u d = d f(d), (2.1.7c) with appropriate initial-boundary conditions. Following the approach of Feireisl [21, 18], we shall obtain the solution of (1.1.1) when ε and δ in (2.1.7). We can solve equation (2.1.7a) provided u is given. Indeed, we can obtain the existence by using classical theory of parabolic 1

18 equation and overcome the difficulty of vacuum. Next we can also solve equation (2.1.7c) when u is fixed. By a direct application of the Schauder fixed point theorem, we can establish the local existence of u, and then extend this local solution to the whole time interval. Note that the addition of the extra term ε u ρ is necessary for keeping the energy conservation. The last step is to let ε and δ to recover the original system. We remark that the strongly nonlinear terms containing d can be controlled by the sufficiently strong estimate about d obtained from the Gagliardo-Nirenberg inequality. In order to control the possible oscillations of the density ρ, we adopt the methods in Lions [39] and Feireisl [21, 18] which is based on the celebrated weak continuity of the effective viscous flux P µdivu. We refer the readers to Lions [39], Feireisl [21, 18], and Hu-Wang [29] for discussions on the effective viscous flux. 2.2 THE SOLVABILITY OF THE DIRECTION VECTOR To solve the approximation system (2.1.7) by the Faedo-Galerkin method, we need to show that the following system can be uniquely solved in terms of u: d t + u d = d f(d), (2.2.1a) d t= = d, d = d, (2.2.1b) which can be achieved by the two lemmas below. Lemma If u C([, T ]; C 2 (, R 3 )), then there exists at most one function d L 2 (, T ; H 2 ()) L ([, T ]; H 1 ()) which solves (2.2.1) in the weak sense on (, T ), and satisfies the initial and boundary conditions in the sense of traces. 11

19 Proof. Let d 1, d 2 be two solutions of (2.2.1) with the same data, then we have (d 1 d 2 ) t + u (d 1 d 2 ) = (d 1 d 2 ) (f(d 1 ) f(d 2 )). (2.2.2) Multiplying (2.2.2) by (d 1 d 2 ), integrating it over, and using integration by parts and the Cauchy-Schwarz inequality, we obtain t (d 1 d 2 ) 2 dx + 2 (d 1 d 2 ) 2 dx = 2 ( (d 1 d 2 )) ( (d 1 d 2 )) udx + 2 C (d 1 d 2 ) 2 dx + (d 1 d 2 ) 2 dx, where we used the fact that f is smooth, then t (d 1 d 2 ) 2 dx + (d 1 d 2 ) 2 dx C (f(d 1 ) f(d 2 ))( (d 1 d 2 ))dx and Lemma follows from Grönwall s inequality, and the above inequality. Lemma Let R 3 be a bounded domain of class C 2+ν, C([, T ]; C 2 (, R 3 )) is a given velocity field. Then the solution operator (d 1 d 2 ) 2 dx, (2.2.3) ν >. Assume that u u d[u] assigns to u C([, T ]; C 2 ( ; R 3 )) the unique solution d of (2.2.1). u d[u] maps bounded sets in C([, T ]; C 2 ( ; R 3 )) into bounded subsets of Moreover, the operator Y := L 2 ([, T ]; H 2 ()) L ([, T ]; H 1 ()), and the mapping u C([, T ]; C 2 ( ; R 3 )) d Y is continuous on any bounded subsets of C([, T ]; C 2 ( ; R 3 )). 12

20 Proof. The uniqueness of the solution to (2.2.1) is a consequence of Lemma 2.2.2, and the existence of a solution can be guaranteed by the standard parabolic equation theory. By (2.2.3), we can conclude that the solution operator u d(u) maps bounded sets in C([, T ]; C( ; 2 R 3 )) into bounded subsets of the set Y. Our next step is to show that the solution operator is continuous from any bounded subset of C([, T ]; C( )) 2 to Y. Let {u n } n=1 be a bounded sequence in C([, T ]; C( )), 2 that is to say, u n B(, R) C([, T ]; C( )) 2 for some R >, and u n u in C([, T ]; C 2 ( )) as n. Here, we denote d[u] = d, and d[u n ] = d n, so we have 1 t 2 (d n d) 2 dx + (d n d) 2 dx = (u d u n d n )( (d n d))dx + (f(d) f(d n )) ( (d n d))dx ( u u n d + u n (d d n ) ) (d n d) dx + C (d n d) 2 dx u n u L d 2 L + C (d d n) 2 2 L + 1 (d 2 n d) 2 dx 2 C u n u L (d d n) 2 L 2, (2.2.4) where we used facts that d n is bounded in Y and f is smooth. This implies that 1 2 t (d n d) 2 dx + 1 (d n d) 2 dx 2 (2.2.5) C u n u L + C (d n d) 2 L 2. Integrating (2.2.5) over time t (, T ), and then taking the upper limit over n on the both sides, we get, noting that u n u in C([, T ]; C 2 ( ); R 3 ), 1 2 lim sup (d n d) 2 dx + 1 n 2 lim n C lim n C T sup lim n T (d n d) 2 L 2dt sup (d n d) 2 L 2dt, 13 T sup (d n d) 2 dxdt (2.2.6)

21 thus, using Grönwall s inequality to (2.2.6) and noting that d n, d share the same initial data, we have which means, from (2.2.6) again, lim n sup (d n d) 2 dx =, lim n T sup (d n d) 2 dxdt =. Thus, we obtain d n d in Y. This completes the proof of the continuity of the solution operator. 2.3 THE FAEDO-GALERKIN APPROXIMATION SCHEME In this section, we establish the existence of solution to the following approximation scheme: ρ t + div(ρu) = ε ρ, (2.3.1a) (ρu) t + div(ρu u) + P (ρ) + δ ρ β + ε u ρ = µ u λdiv ( d d ( 12 ) d 2 + F (d))i 3, (2.3.1b) d t + u d = d f(d), (2.3.1c) with boundary conditions ρ ν =, (2.3.2a) d = d, (2.3.2b) u =, (2.3.2c) 14

22 together with modified initial data ρ t= = ρ,δ (x), (2.3.3a) ρu t= = m,δ (x), (2.3.3b) d t= = d (x). (2.3.3c) Here the initial data ρ,δ (x) C 3 () satisfies the following conditions: < δ ρ,δ (x) δ 1 2β, (2.3.4) and ρ,δ (x) ρ in L γ (), {ρ,δ < ρ } as δ. (2.3.5) Moreover, m if ρ,δ (x) ρ (x), m,δ (x) = if ρ,δ (x) < ρ (x). (2.3.6) The density ρ = ρ[u] is determined uniquely as the solution of the following Neumann initialboundary value problem (see Lemmas 2.1 and 2.2 of [18]): ρ t + div(ρu) = ε ρ, ρ ν =, (2.3.7a) (2.3.7b) ρ t= = ρ,δ (x), (2.3.7c) To solve (2.3.1b) by a modified Faedo-Galerkin method, we need to introduce the finitedimensional space endowed with the L 2 Hilbert space structure: X n = span(η i ) n i=1, n {1, 2, 3, }, where the linearly independent functions η i D() 3, i = 1, 2,..., form a dense subset in C(, 2 R 3 ). The approximate solution u n should be given by the following form: ρu n (τ) ηdx m,δ ηdx τ ( ) = (µ un div(ρu n u n )) (ρ γ + δρ β ) ε ρ u n ηdxdt (2.3.8) τ λdiv( d d ( 1 2 d 2 + F (d))i 3 ) ηdxdt 15

23 for any t [, T ] and any η X n, where ε, δ, β are fixed. Due to Lemmas 2.1 and 2.2 of [18] and our Lemmas and 2.2.2, the problem (2.2.1), (2.3.7) and (2.3.8) can be solved at least on a short time interval (, T n ) with T n T by a standard fixed point theorem on the Banach space C([, T ], X n ). We refer the readers to [18] for more details. Thus we obtain a local solution (ρ n, u n, d n ) in time. To obtain uniform bounds on u n, we derive an energy inequality similar to (2.1.2) as follows. Taking η = u n (t, x) with fixed t in (2.3.1) and repeating the procedure for a priori estimates in Section 2, we deduce a Kinetic energy equality : ( 1 t 2 ρ n u n γ 1 ργ n + δ β 1 ρβ n + λ ) 2 d n 2 + λf (d n ) dx + µ u n 2 dx + λ d n f(d n ) 2 dx + ε (γρ γ 2 n + δβρ β 2 n ) ρ n 2 dx =. (2.3.9) The uniform estimates obtained from (2.3.9) furnish the possibility of repeating the above fixed point argument to extend the local solution u n to the whole time interval [, T ]. Then, by the solvability of equation (2.3.7) and (2.2.1), we obtain the functions (ρ n, d n ) on the whole time interval [, T ]. The next step in the proof of Theorem is to pass the limit as n in the sequence of approximate solutions {ρ n, u n, d n } obtained above. We observe that the terms related to u n and ρ n can be treated similarly to [18]. It remains to show the convergence of the terms related to d n. By (2.3.9), smoothness of f, and elliptic estimates, we conclude u n L 2 ([, T ]; L 2 ()), (2.3.1) d n f(d n ) is bounded in L 2 ([, T ]; L 2 ()), (2.3.11) and d n L ([, T ]; H 1 ()) L 2 ([, T ]; H 2 ()). This yields that d n f(d n ) d f(d) weakly in L 2 ([, T ]; L 2 ()), 16

24 and d n d weakly in L ([, T ]; H 1 ()) L 2 ([, T ]; H 2 ()). (2.3.12) Using corollary 2.1 in [21] and (2.3.1c), we can improve (2.3.12) as follows: d n d in C([, T ]; L 2 weak()). Next we need to rely on the following Aubin-Lions compactness lemma (see [41]): Lemma Let X, X and X 1 be three Banach spaces with X X X 1. Suppose that X is compactly embedded in X and that X is continuously embedded in X 1 ; Suppose also that X and X 1 are reflexive spaces. For 1 < p, q <, let W = {u L p ([, T ]; X ) du dt Lq ([, T ]; X 1 )}. Then the embedding of W into L p ([, T ]; X) is also compact. We are now applying the Aubin-Lions lemma to obtain the convergence of d n and d n. From Remark 2.1.1, we have and Using (2.3.1c), we have d n L 1 ((, T ) ), d n L 1 3 ((, T ) ). (2.3.13) t d n 3 L 2 () C u n d n 3 L 2 () + C d n f(d n ) 3 L 2 () C u n L 6 () d n L 2 () + C d n f(d n ) L 2 (), C u n L 2 () + C d n f(d n ) L 2 (), where we used embedding inequality, the values of C are variant. Thus, (2.3.1), (2.3.11) and (2.3.13) yield t d n L 2 ([,T ];L 2 3 C. ()) Notice that H 2 H 1 L 3 2 and the injection H 2 H 1 is compact, applying Lemma we deduce that the sequence {d n } n=1 is precompact in L 2 ([, T ]; H 1 ()). 17

25 Summing up the previous results, by taking a subsequence if necessary, we can assume that: d n d in C([, T ]; L 2 weak()), d n d weakly in L 2 (, T ; H 2 ()) L (, T ; H 1 ()), d n d strongly in L 2 (, T ; H 1 ()), d n d weakly in L 1 3 ((, T ) ), d n f(d n ) d f(d) weakly in L 2 (, T ; L 2 ()), F (d n ) F (d) strongly in L 2 (, T ; H 1 ()). Now, we consider the convergence of the terms related to d n and d n. Let ϕ be a test function, then ( d n d n d d) ϕdxdt ( d n d n d n d) ϕdxdt + ( d n d d d) ϕdxdt C d n L 2 () d n d L 2 () + C d L 2 () d n d L 2 () By the strong convergence of d n in L 2 () and (2.3.14), we conclude that (2.3.14) d n d n d d in D ( (, T )). Similarly, and 1 2 d n 2 I d 2 I 3 in D ( (, T )), u n d n u d in D ( (, T )), where we used u n u weakly in L 2 ([, T ]; H()). 1 Therefore, (2.2.1) and (2.3.8) hold at least in the sense of distribution. Moreover, by the uniform estimates on u, d and (1.1.1c), we know that the map t d n (x, t)ϕ(x)dx for any ϕ D(), 18

26 is equi-continuous on [, T ]. By the Ascoli-Arzela Theorem, we know that t d(x, t)ϕ(x)dx is continuous for any ϕ D(). Thus, d satisfies the initial condition in (2.2.1). Now we have the existence of a global solution to (2.3.1) as follows: Proposition Assume that R 3 is a bounded domain of the class C 2+ν, ν > ; Let ε >, δ >, and β > max{4, γ} be fixed. Then for any given T >, there is a solution (ρ, u, d) to the initial-boundary value problem of (2.3.1) in the following sense: (1) The density ρ is a nonnegative function such that ρ L γ ([, T ]; W 2,r ()), t ρ L r ((, T ) ), for some r > 1, the velocity u L 2 ([, T ]; H 1 ()), and (2.3.1a) holds almost everywhere on (, T ), and the initial and boundary data on ρ are satisfied in the sense of traces. Moreover, the total mass is conserved, i.e. ρ(x, t)dx = for all t [, T ]; and the following inequalities hold ρ δ, dx, T δ ρ β+1 dxdt C(ε), T ε ρ 2 dxdt C with C independent of ε. (2) All quantities appearing in equation (2.3.1b) are locally integrable, and the equation is satisfied in D ( (, T )). Moreover, 2γ γ+1 ρu C([, T ]; Lweak()), and ρu satisfies the initial data. (3) All terms in (2.3.1c) are locally integrable on (, T ). The direction d satisfies the equation (2.2.1a) and the initial data (2.2.1b) in the sense of distribution. 19

27 (4) The energy inequality ( 1 t 2 ρ u γ 1 ργ + δ β 1 ρβ + λ ) 2 d 2 + λf (d) dx + µ u 2 dx + λ d f(d) 2 dx holds almost everywhere for t [, T ]. To complete our proof of the main theorem, we will take vanishing artificial viscosity and vanishing artificial pressure in the following sections. 2.4 VANISHING VISCOSITY LIMIT In this section, we will pass the limit as ε in the family of approximate solutions (ρ ε, u ε, d ε ) obtained in Proposition The estimates in Proposition are independent of n, and those estimates are still valid for (ρ ε, u ε, d ε ). But, we need to remark that ρ ε will lose some regularity when ε because the term ε ρ ε goes away. The space L (, T ; L 1 ()) is a non-reflexive space, and the artificial pressure is bounded only in space L (, T ; L 1 ()) from the estimates of Proposition It is crucial to establish the strong compactness of the density ρ ε for passing the limits. To this end, we need to obtain better estimates on the artificial pressure Uniform estimates of the density We first introduce an operator B : which is a bounded linear operator satisfying { } f L p () : fdx = [W 1,p ()] 3 B[f] W 1,p () c(p) f L p () for any 1 < p <, (2.4.1) 2

28 where the function W = B[f] R 3 solves the following equation: divw = f in, W =. Moreover, if the function f can be written in the form f = divg for some g L r, and g ν =, then B[f] L r () c(r) g L r () for any 1 < r <. We refer the readers to [21, 18] for more background and discussion of the operator B. Define the function: ϕ(t, x) = ψ(t)b[ρ ε ρ], ψ D(, T ), ψ 1, where ρ = 1 ρ(t)dx. Since ρ ε is a solution to (2.3.1a), by Proposition and β > 4, we have ρ ε ρ C([, T ], L 4 ()). Therefore, from (2.4.1), we have ϕ(t, x) C([, T ], W 1,4 ()). In particular, ϕ(t, x) C([, T ] ) by the Sobolev embedding theorem. Consequently, ϕ can be used as a test function for (2.3.1b). 21

29 After a little bit lengthy but straightforward computation, we obtain: T ψ(ρ γ+1 ε + δρ δ+1 ε )dxdt T T = ρ ψ(ρ γ ε + δρ β ε )dxdt + ψρ ε u ε B[ρ ε ρ]dxdt T + µ ψ u ε B[ρ ε ρ]dxdt T ψρ ε u ε u ε B[ρ ε ρ]dxdt T ε ψρ ε u ε B[ ρ ε ]dxdt T (2.4.2) ψρ ε u ε B[div(ρ ε u ε )]dxdt T + ε u ε ρ ε B[ρ ε ρ]dxdt T λ ( d ε d ε ( d ) ε 2 + F (d))i 3 ψ B[ρ ε ρ]dxdt 2 7 = I j. j=1 To achieve our lemma below, we need to estimate that the terms I 1 I 7 are bounded. We can treat the terms related to ρ ε, u ε similar to [18]. It remains to estimate the term I 7. Indeed, T I 7 = ( d λ ε d ε ( d ) ε 2 + F (d))i 3 ψ B[ρ ε ρ]dxdt 2 T T Cλ d ε 2 B[ρ L 1 ε ρ] 3 () W 1, 5 dt + C B[ρ 2 () ε ρ] W 1, 5 dt 2 () C, (2.4.3) where we used B[ρ ε ρ] W 1, 5 2 () C ρ ε ρ L 5 2 (), and β 4. Consequently, we have proved the following result: Lemma Let (ρ ε, u ε, d ε ) be the solutions of the problem (2.3.1) constructed in Proposition 2.3.1, then ρ ε L γ+1 ((,T ) ) + ρ ε L β+1 ((,T ) ) C, where C is independent of ε. 22

30 2.4.2 The vanishing viscosity limit passage From the previous energy estimates, we have ε ρ ε in L 2 (, T ; W 1,2 ()) and ε u ε ρ ε in L 1 (, T ; L 1 ()) as ε. Due to the above estimates so far, we may now assume that ρ ε ρ in C([, T ], L γ weak ()), (2.4.4a) u ε u weakly in L 2 (, T ; W 1,2 ()), (2.4.4b) γ+1 ρ ε u ε ρu in C([, T ], Lweak ()). 2γ (2.4.4c) Then we can pass the limits of the terms related to ρ ε, u ε similarly to [18]. It remains to show the convergence of d ε. Following the same arguments of Section 4, by taking a subsequence if necessary, we can assume that: d ε d in C([, T ]; L 2 weak()) (2.4.5a) d ε d weakly in L 2 (, T ; H 2 ()) L (, T ; H 1 ()), d ε d strongly in L 2 (, T ; H 1 ()), d ε d weakly in L 1 3 ((, T ) ), d ε f(d ε ) d f(d) weakly in L 2 (, T ; L 2 ()), F (d ε ) F (d) strongly in L 2 (, T ; H 1 ()). (2.4.5b) (2.4.5c) (2.4.5d) (2.4.5e) (2.4.5f) Consequently, letting ε and making use of (2.4.4) and (2.4.5), we conclude that the limit of (ρ ε, u ε, d ε ) satisfies the following system: ρ t + div(ρu) =, (ρu) t + div(ρu u) + P = µ u λdiv( d d ( 1 2 d 2 + F (d))i 3 ), d t + u d = d f(d) (2.4.6a) (2.4.6b) (2.4.6c) where P = aρ γ ε + δρ β ε, here K(x) stands for a weak limit of {K ε }. 23

31 2.4.3 The strong convergence of the density We observe that ρ ε, u ε is a strong solution of parabolic equation (2.3.1a), then the renormalized form can be written as t b(ρ ε ) + div(b(ρ ε )u ε ) + (b (ρ ε )ρ ε b(ρ ε ))divu ε = εdiv(χ b(ρ ε )) εχ b (ρ ε ) ρ ε 2 (2.4.7) in D ((, T ) R 3 ), with b C 2 [, ), b() =, and b, b bounded functions and b convex, where χ is the characteristics function of. By the virtue of (2.4.7) and the convexity of b, we have T ψ(b (ρ ε )ρ ε b(ρ ε )))divu ε dxdt b(ρ,δ )dx + T t ψb(ρ ε )dxdt for any ψ C [, T ], ψ 1, ψ() = 1, ψ(t ) =. Taking b(z) = z log z gives us the following estimate: T ψρ ε divu ε dxdt and letting ε yields T ψρdivudxdt that is, T Meanwhile, (ρ, u) satisfies ρdivudxdt ρ,δ log(ρ,δ )dx + ρ,δ log ρ,δ dx + T T ρ,δ log ρ,δ dx t ψρ ε log ρ ε dxdt, t ψρ log ρdxdt, ρ log ρ(t)dx. (2.4.8) t b(ρ) + div(b(ρ)u) + (b (ρ)ρ b(ρ))divu =. (2.4.9) Using (2.4.9) and b(z) = z log z, we deduce the following inequality: T ρdivudxdt ρ,δ log ρ,δ dx ρ log ρ(t)dx. (2.4.1) From (2.4.1) and (2.4.8), we deduce that ρ log ρ ρ log(ρ)(τ)dx for a.e. τ [, T ]. T ρdivu ρdivudxdt (2.4.11) To obtain the strong convergence of density ρ ε, the crucial point is to get the weak continuity of the viscous pressure, namely: 24

32 Lemma Let (ρ ε, u ε ) be the sequence of approximate solutions constructed in Proposition 2.3.1, then lim T ε + T = ψη(aρ γ ε + δρ β ε µdivu ε )ρ ε dxdt ψη( P µdivu)ρdxdt for any ψ D(, T ), η D(), where P = aρ γ + δρ β. Proof. We need to introduce a new operator A i = 1 ( xi v), i = 1, 2, 3, where 1 stands for the inverse of the Laplace operator on R 3. To be more specific, A i can be expressed by their Fourier symbol A i ( ) = F 1 ( iξ i F( )), i = 1, 2, 3, ξ 2 with the following properties (see [18]): A i v W 1,s () c(s, ) v L s (R 3 ), 1 < s <, A i v L q () c(q, s, ) v L s (R 3 ), q <, provided 1 q 1 s 1 3, and A i v L () c(s, ) v L s (R 3 ) if s > 3. Next, we use the quantities ϕ(t, x) = ψ(t)η(x)a i [ρ ε ], ψ D(, T ), η D(), i = 1, 2, 3, 25

33 as a test function for (2.3.1b) to obtain T ϕη((ρ γ ε + δρ β ε ) µdivu ε )ρ ε dxdt T T = µ ψ u ε ηa[ρ ε ]dxdt ψ(ρ γ ε + δρ β ε ) ηa[ρ ε ]dxdt T T ψρ ε u ε u ε ηa[ρ ε ]dxdt ψ t ηρ ε u ε A[ρ ε ]dxdt T ε ψηρ ε u ε A[div(χ ρ ε )]dxdt T T + ε ψη ρ ε u ε A[ρ ε ]dxdt + µ ψu ε ηρ ε dxdt T T µ ψu ε η A[ρ ε ]dxdt + ψu ε (ρ ε R[ρ ε u ε ] ρ ε u ε R[ρ ε ])dxdt T λ ( d ε d ε ( 12 ) d ε 2 + F (d ε ))I 3 ψ ηa[ρ ε ]dxdt T λ ( d ε d ε ( 12 ) d ε 2 + F (d ε ))I 3 ψη A[ρ ε ]dxdt (2.4.12) where χ is the characteristics function of, A[x] = 1 [x]. Meanwhile, we can use ϕ(t, x) = ψ(t)η(x)( 1 )[ρ], ψ D(, T ), η D(), as a test function for (2.4.6b) to obtain T ϕη( P µdivu)ρdxdt = µ T ψ u ηa[ρ]dxdt T T ψρu u ηa[ρ]dxdt T + µ T + ψu(ρr[ρu] ρur[ρ])dxdt T λ λ T T ψp ηa[ρ]dxdt ψ t ηρua[ρ]dxdt T ψu ηρdxdt µ ψu η A[ρ]dxdt ( d d ( 12 d 2 + F (d))i 3 ) ψ ηa[ρ]dxdt ( d d ( 12 d 2 + F (d))i 3 ) ψη A[ρ]dxdt. (2.4.13) 26

34 For the related terms of ρ ε, u ε, following the same line in [18] we can show that these terms in (2.4.12) converge to their counterparts in (2.4.13). It remains to handle the terms related to d ε in (2.4.12). By virtue of the classical Mikhlin multiplier theorem (see [18]), we have A[ρ ε ] A[ρ] in C([, T ]; L β weak ()) as ε, (2.4.14) and A[ρ ε ] A[ρ] in C((, T ) )) as ε. (2.4.15) Since d ε d ε A[ρ ε ] d da[ρ] dx d ε 2 A[ρ ε ] A[ρ] dx + d ε d ε d A[ρ] dx + d d ε d A[ρ] dx, (2.4.16) using Hölder s inequality to (2.4.16), by (2.4.14), (2.4.15), and (2.4.5c) we have T Similarly, T ( d ε d ε )ψ ηa[ρ ε ]dxdt ( 1 2 d ε 2 I 3 )ψ ηa[ρ ε ]dxdt T T Using the strong convergence of F (d ε ), we conclude that, T λ λ And similarly, T λ λ T T ( d d)ψ ηa[ρ]dxdt as ε. ( 1 2 d 2 I 3 )ψ ηa[ρ]dxdt as ε. ( d ε d ε ( 1 ) 2 d ε 2 + F (d ε ))I 3 ) ψ ηa[ρ ε ]dxdt ( d d ( 1 ) 2 d 2 + F (d))i 3 ) ψ ηa[ρ]dxdt as ε. ( d ε d ε ( 1 ) 2 d ε 2 + F (d ε ))I 3 ) ψη A[ρ ε ]dxdt ( d d ( 1 ) 2 d 2 + F (d))i 3 ) ψη A[ρ]dxdt as ε. 27

35 So we deduce that lim T ε + T = ψη(ρ γ ε + δρ β ε µdivu ε )ρ ε dxdt ψη( P µdivu)ρdxdt for any ψ D(, T ), η D(), where P = ρ γ + δρ β. The proof of Lemma is complete. From Lemma 2.4.2, we have T ρdivu ρdivudxdt 1 µ T ( P ρ aρ γ+1 + δρ β+1 )dxdt. (2.4.17) By(2.4.11) and (2.4.17), we deduce that ρ log(ρ) ρ log(ρ)(τ)dx 1 µ T ( P ρ aρ γ+1 + δρ β+1 )dxdt, and P ρ ρ γ+1 + δρ β+1 due to the convexity of ρ γ + δρ β. So ρ log(ρ) ρ log(ρ)(t)dx. On the other hand, Consequently ρ log(ρ) = ρ log(ρ) that means ρ log(ρ) ρ log(ρ). ρ ε ρ in L 1 ((, T ) ). Thus, we can pass to the limit as ε to obtain the following result: 28

36 Proposition Assume R 3 is a bounded domain of class C 2+ϑ, ϑ >. let δ >, and { } 6γ β > max 4, 2γ 3 be fixed. Then, for any given T >, there exists a finite energy weak solution (ρ, u, d) of the problem: ρ t + div(ρu) =, (2.4.18a) (ρu) t + div(ρu u) + (ρ γ + δρ β ) = µ u λdiv ( d d ( 12 ) d 2 + F (d))i 3, (2.4.18b) d t + u d = d f(d) (2.4.18c) with the boundary condition u =, d = d and initial condition (1.1.2). Moreover, ρ L β+1 ((, T ) ) and the equation (2.4.18a) holds in the sense of renormalized solutions on D ((, T ) R 3 ) provided ρ, u were prolonged to be zero on R 3 \. Furthermore, (ρ, u, d) satisfies the following uniform estimates: sup ρ(t) γ L γ () CE δ[ρ, m, d ], (2.4.19) t [,T ] δ sup ρ(t) β CE L β () δ[ρ, m, d ], (2.4.2) t [,T ] sup ρ(t)u(t) 2 L 2 () CE δ[ρ, m, d ], (2.4.21) t [,T ] u(t) L 2 ([,T ];H 1 ()) CE δ [ρ, m, d ], (2.4.22) sup d 2 L 2 () CE δ[ρ, m, d ], (2.4.23) t [,T ] where C is independent of δ > and E δ [ρ, m, d ] = d L 2 ([,T ];H 2 ()) CE δ [ρ, m, d ], (2.4.24) ( 1 m,δ ρ,δ γ 1 ργ,δ + δ β 1 ρβ,δ + λ ) 2 d 2 + λf (d ) dx. 29

37 Remark Recalling the modified initial data (2.3.3)-(2.3.6), we conclude that the modified energy E δ [ρ, m, d ] is bounded, and consequently the estimates in Proposition hold independently of δ. 2.5 PASSING TO THE LIMIT IN THE ARTIFICIAL PRESSURE TERM The objective of this section is to recover the original system by vanishing the artificial pressure term. Again in this part the crucial issue is to recover the strong convergence for ρ δ in L 1 space Better estimate of density Let us begin with a renormalized continuity equation b(ρ δ ) t + div(b(ρ δ )u δ ) + (b (ρ δ )ρ δ b(ρ δ ))divu δ = in D ((, T ) R 3 ) for any uniformly bounded function b C 1 [, ). We can regularize the above equation as t S m [b(ρ)] + div(s m [b(ρ)]u) + S m [(b (ρ)ρ b(ρ))divu] = q m on (, T ) R 3, (2.5.1) where S m (v) denotes a spatial convolution with a family of regularizing kernels, and q m in L 2 (, T ; L 2 (R 3 )) as m, provided b is uniformly bounded (see details in [18]). We use the operator B to construct multipliers of the form ϕ(t, x) = ψ(t)b[s m [b(ρ δ )] 1 S m [b(ρ δ )]dx], ψ D(, T ), ψ 1, where the operator B was defined in Section 2.4. Taking b(ρ δ ) = ρ σ δ, using (2.5.1) and (2.4.19), with σ small enough, we see that S m [ρ σ δ ] 1 S m [ρ σ δ ]dx 3

38 is in the space C([, T ]; L p ()) for any finite p > 1. By (2.4.1) and the embedding theorem, we have ϕ(t, x) C([, T ] ). Consequently, ϕ(t, x) can be used as a test function for (2.4.18b), then one arrives at the following formula: T T = ψ(ρ γ δ + δρβ δ )S m[ρ σ δ ]dxdt ( ψ(t) (ρ γ δ + δρβ δ )dx)( 1 T i=1 ) S m [ρ σ δ ]dx dt ψ t ρ δ u δ B[S m [ρ σ δ ] 1 S m [ρ σ δ ]dx]dxdt T + ψ(µψ u δ ρ δ u δ u δ ) B[S m [ρ σ δ ] 1 S m [ρ σ δ ]dx]dxdt T + ψρ δ u δ B[S m (ρ σ δ σρ σ δ )divu 1 S m [(ρ σ δ σρ σ δ )divu δ ]dx]dxdt T T ψρ δ u δ B[divS m [(ρ σ δ u δ )]]dxdt + ψρ δ u δ B[q m 1 q m dx]dxdt T + λ ψ ( d δ d δ ( 12 ) d δ 2 + F (d δ ))I 3 B[S m [ρ σ δ ] 1 S m [ρ σ δ ]dx]dxdt 6 T = I i + ψρ δ u δ B[q m 1 q m dx]dxdt. Noting that q m in L 2 (, T ; L 2 (R 3 )) as m, we can pass to the limit for m in the above equality to get the following: T ψ(ρ γ+σ δ + δρ β+σ δ )dxdt Now, we can estimate the integrals I 1 I 6 as follows. (1) We see that I 1 = T 6 I i. i=1 ( ψ(t) (aρ γ δ + δρβ δ )dx)( 1 ) S m (ρ σ δ )dx dt is bounded uniformly in δ provided σ γ by (2.4.19) and (2.4.2). (2) As for the second term, by (2.4.19), (2.4.21), (2.4.22) and together with the embedding W 1,p () L () for p > 3, we have T I 2 = ψ t ρ δ u δ B[S m (ρ σ δ ) 1 S m (ρ σ δ )dx]dxdt c T ψ t dt C 31

39 provided σ γ 3. (3) Similarly, for the third term, we have T I 3 = ψ(µψ u δ ρ δ u δ u δ ) B[S m (ρ σ δ ) 1 C if we choose σ γ 2 ; (4) For I 4, by Hölder inequality, we have T I 4 = ψρ δ u δ B[S m (ρ σ δ σρ σ δ )divu 1 where C T ρ δ L γ () u δ L 6 () ρ θ δdivu δ L q ()dt, p = 6γ 5γ 6, q = 6γ 7γ 6. S m (ρ σ δ )dx]dxdt S m (ρ σ δ σρ σ δ )divu δ dx]dxdt If we choose σ 2γ 3 1, and use (2.4.19), (2.4.2) and (2.4.22), we conclude that I 4 is uniformly bounded. (5) Using the embedding inequality, we have T I 5 = ψρ δ u δ B[divS m (ρ σ δ u δ )]dxdt where r = bounded. C T T 3γ 2γ 3. If we choose σ 2γ 3 (6) Finally, we estimate term I 6, let σ γ 2, then ρ δ L γ u δ L 6 ρ σ δ u δ L pdt ρ δ L γ u δ 2 L 6 ρσ δ L rdt, 1, and use (2.4.19), (2.4.2) and (2.4.22), then I 5 is I 6 T = λ ψ ( d δ d δ ( 12 ) d δ 2 + F (d δ ))I 3 B[S m (ρ σ δ ) 1 T C d δ 2 B[S m(ρ σ L 1 3 () δ ) 1 S m (ρ σ δ )dx] 5 L 2 () dt T + C B[S m (ρ σ δ ) 1 S m (ρ σ δ )dx] 5 L 2 () dt C, S m (ρ σ δ )dx]dxdt 32

40 where we used the smoothness of F, (2.4.1), (2.4.19), (2.4.2) and d δ L 1 3 ((, T ) ). All those above estimates together yield the following lemma: Lemma Let γ > 3. There exists σ > depending only on γ, such that 2 ρ γ+σ δ + δρ β+σ δ is bounded in L 1 ((, T ) ) The limit passage By virtue of the estimates in Proposition and Remark 2.4.1, we can assume that, up to a subsequence if necessary, ρ δ ρ in C([, T ], L γ weak ()), (2.5.2) u δ u weakly in L 2 ([, T ]; H 1 ()), (2.5.3) d δ d weakly in L 2 ([, T ]; H 2 ()) L ([, T ]; H 1 ()). (2.5.4) d δ d strongly in L 2 (, T ; H 1 ()), (2.5.5) d δ d weakly in L 1 3 ((, T ) ), (2.5.6) d δ f(d δ ) d f(d) weakly in L 2 (, T ; L 2 ()), (2.5.7) Letting δ, we have, F (d δ ) F (d) strongly in L 2 (, T ; H 1 ()). (2.5.8) ρ γ δ ργ weakly in L 1 ((, T ) ()), (2.5.9) 33

41 subject to a subsequence. From (2.5.5) and (2.5.8), we have, as δ, d δ d δ ( 1 2 d δ 2 + F (d δ ))I 3 d d ( 1 2 d 2 + F (d))i 3 in D ( (, T )), (2.5.1) and u δ d δ u d in D ( (, T )), (2.5.11) as δ. On the other hand, by virtue of (2.4.18b), (2.4.19)-(2.4.22), we obtain ρ δ u δ ρu in C([, T ]; L 2γ γ+1 weak ()). (2.5.12) Similarly, we have, as δ, d δ d in C([, T ]; L 2 weak()). By Lemma 2.5.1, we get δρ β δ in L1 ((, T ) ) as δ. Thus, the limit of (ρ, ρu, d) satisfies the initial and boundary conditions of (1.1.2) and (1.1.3). Since γ > 3, (2.5.3) and (2.5.12) combined with the compactness of 2 H1 () L 2 () imply, as δ, ρ δ u δ u δ ρu u in D ((, T ) ). Consequently, letting δ in (2.4.18) and making use of (2.5.2)-(2.5.12), the limit of (ρ δ, u δ, d δ ) satisfies the following system: ρ t + div(ρu) =, (ρu) t + div(ρu u) + ρ γ = µ u λdiv ( d d ( 12 ) d 2 + F (d))i 3 d t + u d = d f(d) (2.5.13a) (2.5.13b) (2.5.13c) in D ( (, T )). 34

42 2.5.3 The strong convergence of density In order to complete the proof of Theorem 2.1.1, we still need to show the strong convergence of ρ δ in L 1 (), or, equivalently ρ γ = ρ γ. Since ρ δ, u δ is a renormalized solution of the equation (2.5.13a) in D ((, T ) R 3 ), we have T k (ρ δ ) t + div(t k (ρ δ u δ )) + (T k (ρ δ )ρ δ T k (ρ δ ))div(u δ ) = in D ((, T ) R 3 ), where T k (z) = kt ( z ) for z R, k k = 1, 2, 3 and T C (R) is chosen so that T (z) = z for z 1, T (z) = 2 for z 3, T convex. Passing to the limit for δ we deduce that t T k (ρ) + div((t k (ρ))u) + (T k (ρ)ρ T k(ρ))divu = in D ((, T ) R 3 )), where T k(ρδ)ρ δ T k (ρ δ )divu δ (T k (ρ)ρ T k(ρ))divu weakly in L 2 ((, T ) ), and T k (ρ δ ) T k (ρ) in C([, T ]; L p weak ()) for all 1 p <. Using the function ϕ(t, x) = ψ(t)η(x)a i [T k (ρ δ )], ψ D[, T ], η D(), as a test function for (2.4.18b), by a similar calculation to the previous sections, we can deduce the following result: Lemma Let (ρ δ, u δ ) be the sequence of approximate solutions constructed in Proposition 2.4.1, then T T lim ψη(ρ γ δ µdivu δ)t k (ρ δ )dxdt = ψη(ρ γ µdivu)t k (ρ)dxdt δ for any ψ D(, T ), η D(). 35

43 In order to get the strong convergence of ρ δ, we need to define the oscillation defect measure as follows: T OSC γ+1 [ρ δ ρ]((, T ) ) = sup lim sup T k (ρ δ ) T k (ρ)) γ+1 dxdt. k 1 δ Here we state a lemma about the oscillation defect measure: Lemma There exists a constant C independent of k such that OSC γ+1 [ρ δ ρ]((, T ) ) C for any k 1. Proof. Following the line of argument presented in [18], and by Lemma 2.5.2, we obtain On the other hand, T OSC γ+1 [ρ δ ρ]((, T ) ) lim divu δ T k (ρ δ ) divut k (ρ)dxdt. δ So we can conclude the Lemma. T lim divu δ T k (ρ δ ) divut k (ρ)dxdt δ T = lim (T k (ρ δ ) T k (ρ) + T k (ρ) T k (ρ))divu δ dxdt δ 2 sup u δ L 2 ((,T ) ) lim sup T k (ρ δ ) T k (ρ) L 2 ((,T δ δ ) ). We are now ready to show the strong convergence of the density. To this end, we introduce a sequence of functions L k C 1 (R) : zlnz, z < k L k (z) = zln(k) + z z T k (s) ds, z k. s 2 k Noting that L k can be written as L k (z) = β k z + b k z, where b k satisfy (2.1.6), we deduce that t L k (ρ δ ) + div(l k (ρ δ )u δ ) + T k (ρ δ )divu δ =, (2.5.14) 36

44 and t L k (ρ) + div(l k (ρ)u) + T k (ρ)divu = (2.5.15) in D ((, T ) ). Letting δ, we can assume that L k (ρ δ ) L k (ρ) in C([, T ]; L γ weak ()). Taking the difference of (2.5.14) and (2.5.15), and integrating with respect to time t, we obtain (L k (ρ δ ) L k (ρ))φdx T ( ) = (L k (ρ δ )u δ L k (ρ)u) φ + (T k (ρ)divu T k (ρ δ )divu δ )φ dxdt, for any φ D(). Following the line of argument in [18], we get ( ) L k (ρ) L k (ρ) (t)dx T T = T k (ρ)divudxdt lim T k (ρ δ )divu δ dxdt. δ + (2.5.16) We observe that the term L k (ρ) L k (ρ) is bounded by its definition. Using Lemma and the monotonicity of the pressure, we can estimate the right-hand side of (2.5.16): T T T T k (ρ)divudxdt lim δ + (T k (ρ) T k (ρ))divudxdt. T k (ρ δ )divu δ dxdt (2.5.17) By virtue of Lemma 2.5.3, the right-hand side of (2.5.17) tends to zero as k. So we conclude that ρ log(ρ)(t) = ρ log(ρ)(t) as k. Thus we obtain the strong convergence of ρ δ in L 1 ((, T ) ). Therefore we complete the proof of Theorem