On the Doi Model for the suspensions of rod-like molecules in compressible fluids

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1 On the Doi Moel for the suspensions of ro-like molecules in compressible fluis Hantaek Bae an Konstantina Trivisa Key wors: Doi moel, suspensions of ro-like molecules, flui-particle interaction moel, compressible Navier-Stokes equations, Fokker-Planck-type equation. Abstract Polymeric fluis arise in many practical applications in biotechnology, meicine, chemistry, inustrial processes an atmospheric sciences. In this article, the Doi moel for the suspensions of ro-like molecules in a compressible flui is investigate. The moel uner consieration couples a Fokker-Planck-type equation on the sphere for the orientation istribution of the ros to the Navier-Stokes equations for compressible fluis, which are now enhance by aitional stresses reflecting the orientation of the ros on the molecular level. The couple problem is 5-imensional (three-imensions in physical space an two egrees of freeom on the sphere) an it escribes the interaction between the orientation of ro-like polymer molecules on the microscopic scale an the macroscopic properties of the flui in which these molecules are containe. Prescribing arbitrarily the initial ensity of the flui, the initial velocity, an the initial orientation istribution in suitable spaces we establish the global-in-time existence of a weak solution to our moel efine on a boune omain in the three imensional space. The proof relies on the construction of a sequence of approximate problems by introucing appropriate regularization an the establishment of compactness. Contents 1 Introuction 2 2 Definition of weak solution an main results A priori estimate Definition of weak solution, main result Proof of Proposition 2.3 (i), (ii), (iii), formal proof of (v), an (vi) Proof of Proposition 2.3. (i) Proof of Proposition 2.3. (ii) Proof of Proposition 2.3. (iii) Proof of Proposition 2.3 (v): formal proof on the whole spaces Proof of Proposition 2.3 (vi) Center for Scientific Computation an Mathematical Moeling, University of Marylan, College Park, MD hbae@cscamm.um.eu. Department of Mathematics & Institute for Physical Science an Technology, University of Marylan, College Park, MD trivisa@math.um.eu 1

2 4 Proof of Proposition 2.3 (iv) an (v) Higher integrability of ρ Limit of the effective viscous flux Proof of Proposition 2.3. (iv) Proof of Proposition 2.3. (v) Construction of approximate sequences Smoothing ρ t ρu Nonlinear amping Truncation of the pressure Appenix Appenix 1: erivation of the equation of ψ Appenix 2: verification of the formal proof in Section References 3 1 Introuction The evolution of ro-like molecules in both compressible an incompressible fluis is of great scientific interest with a variety of applications in science an engineering. The present article eals with the Doi moel for the suspensions of ro-like molecules in a ilute regime. The moel uner consieration couples a Fokker-Planck-type equation on the sphere for the orientation istribution of the ros to the Navier-Stokes equations for compressible fluis, which are now enhance by aitional stresses reflecting the orientation of the ros on the molecular level. The couple problem is 5-imensional (three-imensions in physical space an two egrees of freeom on the sphere) an it escribes the interaction between the orientation of ro-like polymer molecules on the microscopic scale an the macroscopic properties of the flui in which these molecules are containe. The macroscopic flow leas to a change of the orientation an, in the case of flexible particles, to a change in shape of the suspene microstructure. This process, in turn yiels the prouction of a flui stress. In this paper, we consier the Doi moel for a compressible flui in a boune omain. The erivation of the system uner consieration is escribe below. A smooth motion of a boy in continuum mechanics is escribe by a family of one-to-one mappings X(t, ) :, t I. The curve X(t, x) represents the trajectory of a particle occupying at time t a spatial position x an this curve is completely etermine by a velocity fiel u : I R 3 through X(t, x) = u(t, X(t, x)), X(, a) = a. t Then, the conservation of mass can be formulate as follows: ρ(t, x)x =, B, t X(t,B) 2

3 where ρ is a nonnegative function that correspons to the ensity of the flui. This equation is equivalent to ρ(t, x)x ρ(t, x)u(t, x) ˆnS =, t B B where ˆn is the unit outer normal vector on. If ρ is smooth, one can use Green s theorem to euce the following continuity equation: ρ t (uρ) =. (1.1) We next obtain equation of motion by applying Newton s secon law of motion as follows: ρ(t, x)u(t, x)x = ρ(t, x)f (t, x)x t(t, x, ˆn)S. t X(t,B) Then, we have ρ(t, x)u(u, x)x t B B X(t,B) (ρu)(t, x)u(t, x) ˆnS = B X(t,B) ρ(t, x)f (t, x)x t(t, x, ˆn)S. (1.2) B For the simplicity, we take F =. The stress principle in continuum mechanics can be aresse through the funamental laws of Cauchy: there is a symmetric stress tensor T(t, x) such that Therefore, (1.2) becomes ρ(t, x)u(u, x)x t B t(t, x, ˆn) = T(t, x)ˆn. By applying Green s lemma to (1.3), we finally have B (ρu)(t, x)u(t, x) ˆnS = (ρu) t (ρu u) = T, ( T) i = The stress tensor T of a general flui obeys Stokes law: T = S pi 3 3, 3 j=1 B T(t, x)ˆns. (1.3) T ij x j. (1.4) where p is the pressure an S is the stress tensor. Let us etermine S an p in our moel. The pressure p is of the form S consists of two parts: p = aρ γ, γ > 3 2. (1.5) S = S 1, where S 1 is the viscous stress tensor generate by the flui S 1 = µ ( u ( u) t) λ( u)i 3 3, 3

4 an is the macroscopic symmetric stress tensor erive from the orientation of the ros at the molecular level. The microscopic insertions at time t an macroscopic place x are escribe by the probability f(t, x, τ)τ. The suspension stress tensor is given by an expansion where an (x, t) = σ (1) (x, t) σ (2) (x, t) σ (3) (x, t), σ (1) (t, x) = (3τ τ I 3 3 )f(t, x, τ)τ, σ (2) (t, x) = σ (2) ij (t, x)i 3 3, with σ (2) ij (t, x) = σ (3) (t, x) = σ (3) ij (t, x)i 3 3, with σ (3) ij (t, x) = γ (2) ij (τ)f(t, x, τ)τ, γ (3) ij (τ 1, τ 2 )f(t, x, τ 1 )f(t, x, τ 2 )τ 1 τ 2. This, an more general expansions for are encountere in the polymer literature (cf. Doi an Ewars 8). We refer the reaer to the articles by Constantin et al 5, 6, where a general class of stress tensors is presente in the context of incompressible fluis. The structure coefficients in the expansion γ (2) ij, γ(3) ij are in general smooth, time inepenent, x inepenent, an o not epen on f. Assuming for simplicity that γ (2) ij (τ) = γ(3) ij (τ 1, τ 2 ) = 1 an enoting η(t, x) = f(t, x, τ)τ the suspension stress tensor takes the form (x, t) = σ (1) (x, t) ηi 3 3 η 2 I 3 3. (1.6) In this setting, f escribes the time-epenent orientation istribution that a ro with a center mass at x has an axis τ in the area element τ an it is escribe by a compressible Fokker-Plank type equation, f t (uf) τ (P τ uτf) D τ τ f D f =, (1.7) where P τ ( x uτ) = x uτ (τ x uτ)τ is the projection of uτ on the tangent space of at τ. With τ an τ we enote the graient an the Laplace operator on the unit sphere, while an represent the graient an the Laplacian operator in R 3. The secon term (uf) in (1.7) escribes the change of f ue to the isplacement of the center of mass of the ros ue to macroscopic avection. The term τ (P τ uτf) is a rift-term on the sphere, which represents the shear-forces acting on the ros. The term D τ τ f represents the rotational iffusion ue to Brownian motion. This effect causes the ros to change their orientation spontaneously, whereas the term D f is the translational iffusion ue to Brownian effects. By integrating (1.7) over, we can obtain the equation of η: η t (uη) D η =. (1.8) 4

5 By substituting (1.6) an (1.5) to (1.4), the equation of motion becomes (ρu) t (ρu u) µ u λ ( u) a ρ γ η 2 = σ η. (1.9) In sum, after normalizing all the constants by 1 for the sake of simplicity, we have the following system of equations: ρ t (ρu) = in (, T ), (1.1a) (ρu) t (ρu u) u ( u) ρ γ η 2 = σ η in (, T ), (1.1b) f t (uf) τ (P τ ( x uτ)f) τ f x f = in (, T ), (1.1c) where is a boune omain an we impose Dirichlet bounary conitions to u, f, an η: u =, f =, an η =, on. In the sequel, we construct a sequence of approximating problems by regularizing the equations by extening functions to be zero outsie. Prescribing arbitrarily the initial flui ensity, the initial velocity, an the initial orientation istribution in suitable spaces, we establish long-time an large ata existence of a weak solution. Since the efinition of a weak solution an the main result are rather complicate, they are state in Section 2. Relate results on the Doi moel for the suspensions of ro-like molecules in incompressible fluis have been stuie by many authors. We refer the reaer to Constantin 5, 6, 7, Lions an Masmoui 14, 15, Masmoui 16 an Otto an Tzavaras 19 for results on relate moels on the whole space. In 1 the authors treat the Doi moel for an incompressible flui within a boune omain in the 3-imensional space an establish results on the global existence of solutions. For compressible moels, relate results have been presente in a series of articles. We refer the reaer to Carrillo et al 2, 3, 4, Gouon et al 11, 11, 12, an Mellet an Vasseur 17, 18, where asymptotic, analytical an numerical results on relate flui-particle interaction moels are iscusse. These articles eal with moels coupling the Stokes, Navier Stokes or Euler system with either the Smolukowski equation or Fokker-Planck equation. What istinguishes the moel presente in this article, besies the general type of the stress tensor uner consieration, is the fact that, unlike other moels, the Fokker-Planck-type equation presente here takes into consieration in aition to the Brownian effects the presence of the shear forces acting on the ros. This new element yiels a new equation for the entropy inuce by the probability ensity function f in the microscopic level an therefore new apriori estimates. We refer the reaer to Appenix 1 for the erivation of equation of the entropy ψ. The paper is organize as follows. In Section 2, we introuce the notion of a weak solution of the system (1.1), an we state the main results; compactness (Proposition 2.3) an existence (Theorem 2.2). In Section 3, we prove various convergence results an we provie a formal proof of the strong convergence of ρ uner better assumptions. This formal convergence argument is recaste in Section 4, where the strong convergence of ρ is prove by using suitable cut-off functions in the renormalize equation of ρ. In Section 5, we generate an approximate sequence of weak solutions in three steps. (i) We first regularize the equation of ρ, which correspons to the regularization of ρ t ρu in (1.1b). We also regularize u in the equation of f which requires to regularize η an σ in the right-han sie of (1.1b). Before regularization, we exten equations to zero outsie. (ii) Next we a nonlinear amping terms to the equation of ρ an η to increase integrability of ρ an η. (iii) We finally truncate ρ γ an η 2 to increase regularity of {ρ, u, η}. By passing to the 5

6 limits in sequences, we can prove Theorem 2.2. Notations: L p (, T ; X) enotes the Banach set of Bochner measurable functions f from (, T ) ( ) 1 T to X enowe with either the norm g(, t) p X t p for 1 p < or sup g(, t) X for p =. t> ( In particular, f L r T (, T ; XY ) enotes ( ) ) 1 f(t) Yτ p X t p or sup ( ) f(t) X Yτ for p =. t> A B means there is a constant C such that A CB. X comp Y means that X is compactly embee in Y. I X is the inicator function which is 1 for x X an otherwise. C(T ) is a function only epening on initial ata an T. an enote weak limit an strong limit, respectively. 2 Definition of weak solution an main results 2.1 A priori estimate Before introucing the concept of a weak solution of the system (1.1), let us present the energy estimate. Multiplying (1.1b) by u an integrating over we get ρ u 2 ργ t 2 γ 1 η2 x u 2 u 2 2 η 2 x (2.1) = u : σx ( u)ηx. Next we introuce an entropy inuce by f in the microscopic level. Let ψ(t, x) = (f ln f)(t, x, τ)τ. Then, ψ satisfies ψ t (uψ) ψ 4 τ f 2 τ 4 f 2 τ = u : σ ( u)η. (2.2) For the erivation of (2.2), we refer the reaer to Section 6.1. Integrating (2.2) over, we obtain ψx 4 τ f 2 τx 4 f 2 τx = u : σx ( u)ηx. (2.3) t By aing (2.3) to (2.1), we have ρ u 2 t 2 ργ γ 1 η2 ψ x 4 u 2 u 2 2 η 2 x =. τ f 2 τx 4 f 2 τx In particular, η is boune in L (, T ; L 2 ()) L 2 (, T ; H 1 ()), which cannot be obtaine erive from (1.8) in the three imensions. From (2.4), we can obtain various estimates of {ρ, u, f, η, σ}. First, ρ u 2 L (, T ; L 1 ()), ρ L (, T ; L γ ()), u L 2 (, T ; L 2 ()), ψ L (, T ; L 1 ()), η L (, T ; L 2 ()) L 2 (, T ; H 1 ()). 6 (2.4) (2.5)

7 By expressing ρu as ρ ρu, we have ρu L (, T ; L 2γ γ1 ()). (2.6) From the entropy issipation, f L 2 (, T ; L 2 ()H 1 ( ) H 1 ()L 2 ( ) ) L 2(, T ; L 2 ()L 6 ( ) L 6 ()L 2 ( ) ), which implies that f L 1(, T ; L 1 ()L 3 ( ) L 3 ()L 1 ( ) ) L 1 (, T ; L 2 ( )). (2.7) Since σ(t, x) 3 f(t, x, τ)τ = 3η(t, x), σ L 1 (, T ; L 3 ()) L (, T ; L 2 ()). (2.8) We next estimate the erivative of σ by using the entropy issipation. σ(t, x) 3 f(t, x, τ) τ 1 f 2 2 τ ( 1 f) 2 2 τ = 1 f 2 2 τ η 1 2. Since η 1 2 L (, T ; L 4 ()) L 2 (, T ; L 6 ()), σ L 1 (, T ; L 3 2 ()) L 2 (, T ; L 4 3 ()). (2.9) 2.2 Definition of weak solution, main result We now efine a weak solution of the system (1.1). By a notational abuse, we inclue η an σ in the efinition of weak solution. Definition 2.1 We say that {ρ, u, f, η, σ} is a weak solution of the system (1.1) if (i) (1.1a) hols in the sense of renormalize solutions, i.e., b(ρ) t (b(ρ)u) (b (ρ)ρ b(ρ)) u = (2.1) for any b C 1 such that b (z)z b(z) C for all z R. (ii) (1.1b) an (1.1c) hol in the sense of istributions. (iii) Moreover, {ρ, u, f, η, σ} satisfies the following energy inequality: ρ u 2 ργ 2 γ 1 η2 ψ (t)x 4 t 4 t f 2 τxt ρ u 2 ργ 2 γ 1 η2 ψ x t τ f 2 τxt S 2 u 2 u 2 2 η 2 xt (2.11) 7

8 Remark 1 (1) The central ifficulty in showing the existence of a weak solution in the theory of compressible fluis is typically the epenence of the pressure on nonlinear terms, for instance ρ γ. From the a priori estimate, we have ρ L (, T ; L γ ()), which is not enough to pass to the limit to ρ γ in the sense of istributions. The issue is resolve by showing that ρ satisfies a better integrability conition by choosing appropriate cut-off functions in the renormalize form (2.1) in the spirit of Feireisl 9 (see also Lions 13). Note that in the present context the suspension stress tensor epens on the ensity of the particles in a nonlinear way as well. In this case, the regularity of η, η L 2 (, T ; H 1 ()) enables us to hanle the nonlinearity. (2) The main aitional ifficulties in the present context involve the presence of two nonlinear terms in the equation of f. In fact, letting χ Cc ( ) we obtain from the avection term (uf), (u (n) f (n) )χτx = u (n) i xi χf (n) τ x, (2.12) whereas from the shear term τ (P τ ( x uτ)f), we get τ (P τ ( x u (n) τ)f (n) )χτx = To pass to the limit in (2.12) an (2.13), we nee to show that xi χf (n) (n) χ τ, τ j f τ τ i u (n) i (n) χ τ j f τ x. (2.13) x j τ i converge strongly in L 2 (, T ; L 2 ()). This is prove in Section 3. (3) In orer to pass to the limit to linear terms in the equation of f, we only nee f L p (, T ; L q ( )) for some p > 1 an q > 1. Since f L (, T ; L 1 ( )) L 1 (, T ; L 2 ( )), we can choose, for example, f L 2 (, T ; L 6 5 ( )). We now state the main result of the paper. Theorem 2.2 Let γ > 3 2 an be a C1 boune omain. Assume that a sequence {ρ, u, f, η } satisfies ρ L 1 L γ (), m 2 ρ L 1 () for ρ, ρ u = m L 2γ γ1 (), f L 1 ( ), η L 2 (), m 2 ρ = for ρ =, (2.14) Then, there exists a weak solution {ρ, u, f, η, σ} of the system (1.1) satisfying (2.14) at t =. Moreover, ρ L p ( (, T )), p = 5 γ 1. (2.15) 3 The proof of Theorem 2.2 consists of two parts. First, we prove the compactness of an approximate sequence {ρ n, u n, f n, η n, σ n } n 1 uner the assumption γ > 3 2 in Section 3 an 4. We state the etaile statement in Proposition 2.3 below. Seconly, we construct an approximate sequence of solutions through regularizing equations in Section 5. The reaer shoul contrast the approximating scheme presente here with the schemes presente in 4, 1 an 13 for ifferent moels. 8

9 We begin with the compactness result. Suppose there is a an approximate sequence of solutions {ρ n, u n, f n, η n, σ n } n 1 such that {ρ n } is boune in L (, T ; L 1 L γ ()), {ρ n u n 2 } is boune in L (, T ; L 1 ()), {u n } is boune in L 2 (, T ; H 1 ()), {f n } is boune in L 2 (, T ; L 6 5 ( )), {η n } is boune in L (, T ; L 2 ()) L 2 (, T ; H 1 ()), {σ n } is boune in L (, T ; L 2 ()) L 1 (, T ; L 3 ()), { σ n } is boune in L 1 (, T ; L 3 2 ()) L 2 (, T ; L 4 3 ()). (2.16) Then, we can extract a subsequence, using the same notation, {ρ n, u n, f n, η n, σ n } n 1 such that ρ n ρ in L γ ( (, T )) an ρ L (, T ; L 1 L γ ()), ρ n u n v in L 2 (, T ; L 2 ()) an v L (, T ; L 2 ()), u n u in L 2 (, T ; H 1 ()), ρ n ρ in L 2γ ( (, T )), ρ n u n m in L 2γ γ1 ( (, T )) an m L (, T ; L 2γ γ1 ()) ρ n u n i u n j e ij in the sense of measures an e ij is a boune measure, f n f in L 2 (, T ; L 6 5 ( )), η n η in L 2 (, T ; H 1 ()) an η L (, T ; L 2 ()) L 2 (, T ; H 1 ()), σ n σ in L 2 (, T ; L 2 ()) an σ L (, T ; L 2 ()) L 1 (, T ; L 3 ()), σ n σ in L 2 (, T ; L 4 3 ()) an σ L 1 (, T ; L 3 2 ()) L 2 (, T ; L 4 3 ()). (2.17) Proposition 2.3 (Compactness) Let γ > 3 2 an be a C1 boune omain. Assume that the energy inequality (2.11) hols for a sequence {ρ n, u n, f n, η n, σ n } n 1. Then, limit functions in (2.17) satisfy the followings. (i) v = ρu, m = ρu, e ij = ρu i u j. (ii) η n converges strongly to η in L 2 ( (, T )), an σ n converges strongly to σ in L 2 ( (, T )). (iii) ρ n (η n ) 2 converges to ρη 2 in the sense of istributions. (iv) ρ an u solve (1.1a) in the sense of renormalize solutions. (v) If in aition we assume that ρ n converges to ρ in L 1 (), then {ρ, u, f, η, σ} is a weak solution of (1.1) such that ρ n ρ in L 1 ( (, T )) C(, T ; L p ()) for all 1 p < γ. (2.18) (vi) Finally, we have the following strong convergence: ρ n u n ρu in L p (, T ; L r ()) for all 1 p <, 1 r < 2γ γ 1, u n u in L p ( (, T )) {ρ > } for all 1 p < 2, u n u in L 2 ( (, T )) {ρ δ} for all δ >, ρ n u n i u n j ρu i u j in L p (, T ; L 1 ()) for all 1 p <. (2.19) We will prove this proposition in Section 3 an 4. 9

10 3 Proof of Proposition 2.3 (i), (ii), (iii), formal proof of (v), an (vi) 3.1 Proof of Proposition 2.3. (i) We begin with the proof of Proposition 2.3 (i). To this en, we nee the following lemma. Lemma Let g n an h n converge weakly to g an h respectively in L p 1 (, T ; L p 2 ()) an L q 1 (, T ; L q 2 ()), where 1 p 1, p 2, 1 p 1 1 q 1 = 1 p 2 1 q 2 = 1. Suppose t gn is boune in L 1 (, T ; W m,1 ()) for some m inepenent of n, h n h n ( ξ, t) L q 1 (,T ;L q 2 ()) as ξ, uniformly in n. Then, g n h n converges to gh in the sense of istributions. We woul like to apply Lemma 3.1 to h n = u n with q 1 = 2 an q 2 2, 6), an g n = ρ n, ρ n u n, or g n = ρ n. First, we nee to show that {ρ n } is boune in L 2 (, T ; L p ), p > 6 5. But, this is clear because ρ n L (, T ; L γ ()), γ > 3 2 > 6 5. Next, from the equation of ρ, (ρ n ) t = (ρ n u n ) is boune in L 2γ 1, (, T ; W γ1 ()) L 1 (, T ; W 1,1 ()). Therefore, m = ρu. Since u n L 2 (, T ; L 6 ()), ρ n u n L 2 (, T ; L p ()), Next, from the equation of motion, 1 p = 1 γ 1 6 < = 5 6. (ρ n u n ) t = (ρ n u n u n ) u n u n (ρ n ) γ (η n ) 2 σ n η n L (, T ; W 1,1 ()) L 2 (, T ; H 1 ()) L (, T ; W 1,1 ()) is boune in L 1 (, T ; W 1,1 ()). Here, we use the fact that (η n ) 2, σ n, an η n are boune in L (, T ; L 1 ()). Therefore, e = m u = ρu u. ρ n satisfies that ρ t n (u n ρ n ) = 1 2 ( un ) ρ n, from which t ρ n is boune in L (, T ; W 1,2 ()) L 2 (, T ; L 2γ γ1 ()) L 1 (, T ; W 1,1 ()). Since ρ n L (, T ; L 2γ ()), 2γ > 6 5, we conclue as above that v = ρu. 3.2 Proof of Proposition 2.3. (ii) To show the strong convergence of η n an σ n, we nee the following lemma. Lemma Let X, B, an Y be Banach spaces such that X is compactly embee in B an B is a subset of Y. Then, for 1 p <, {v; v L p (, T ; X), v t L 1 (, T ; Y )} is compactly embee in L p (, T ; B). (3.1) 1

11 Strong convergence of η n : First, η n t = (u n η n ) η n L 1 (, T ; W 1,1 ()) L 2 (, T ; W 1,2 ()) L 1 (, T ; W 1,1 ()). Since H 1 () comp L 2 () W 1,1 (), η n η in L 2 (, T ; L 2 ()). (3.2) Strong convergence of σ n : First, σ t = (3τ τ I 3 3 )f t (t, x, τ)τ = (3τ τ I 3 3 ) (uf) τ (P τ ( x uτ)f) τ f f τ = (uσ) τ (3τ τ I 3 3 ) ( uτ)fτ τ (3τ τ I 3 3 ) τ fτ σ (uσ) u η τ f τ σ L 1 (, T ; W 1,1 ()) L 1 (, T ; L 1 ()) L 1 (, T ; L 3 2 ()) L 1 (, T ; W 1, 3 2 ()) L 1 (, T ; W 1,1 ()). From σ L 1 (, T ; L 3 ()) L 2 (, T ; L 4 3 ()), we have, for example, σ L 3 2 (, T ; L ()). From 18 1, W 11 () comp L 2 () W 1,1 (), we have σ n σ in L 3 2 (, T ; L 2 ()). Since {σ n } is uniformly boune in L (, T ; L 2 ()), σ n σ in L p (, T ; L 2 ()) for all p <. Therefore, σ n σ in L 2 (, T ; L 2 ()). (3.3) Strong convergence of xi χf (n) (n) χ τ, τ j f τ: We note that these two terms are of τ i the form ςf n τ, where ς Cc ( ), an ςf n τ an their time an spatial erivatives satisfy the same boun of σ. Therefore, these two terms converge strongly in L 2 (, T ; L 2 ()) as well. 3.3 Proof of Proposition 2.3. (iii) To show (iii), we nee to show (η n ) 2 converges strongly in L p (, T ; L q ()) for some p > 1 an q such that 1 q 1 γ 1. Given γ > 3 2, we take ɛ > such that 1 γ = 2 3 ɛ. Then, for θ = 1 ɛ, η n η L p (,T ;L q ()) η n η 1 ɛ L 2 (,T ;L 6 δ ()) ηn η ɛ L (,T ;L 2 ()), where 1 p = 1 ɛ 2, 1 q = 1 ɛ 6 2δ ɛ 2. 11

12 Since H 1 () comp L 6 2δ () for any δ >, Let p = p 2 an q = q 2. Then, η n η in L p (, T ; L q ()). (η n ) 2 η 2 in L p (, T ; L q ()), (3.4) where 1 q 1 γ 1 by taking δ < 3ɛ. Therefore, ρn (η n ) 2 converges to ρη 2 in the sense of istributions. 3.4 Proof of Proposition 2.3 (v): formal proof on the whole spaces Before proving Proposition 2.3 (iv) an (v), we provie a formal proof of the convergence of ρ n on the whole spaces uner a stronger assumption: {ρ n } is boune in L γ1 ((, T ) R 3 ) L (, T ; L s (R 3 )), s > 3. (3.5) For etails of the proof, see Section 6.2. From (1.1a), Next, we take ( ) 1 to (1.1b). Then, (ρ log ρ) t (uρ log ρ) ( u)ρ =. (3.6) ( ) 1 (ρu) ( ) 1 i j (ρu i u j ) 2 u ρ γ η 2 = ( ) 1 ( σ η), t from which we have 2 u = ( ) 1 (ρu) ( ) 1 i j (ρu i u j ) ρ γ η 2 ( ) 1 ( σ η). (3.7) t By (3.6) an (3.7), 2 (ρ log ρ) t (uρ log ρ) ρ γ1 = ρ ( ) 1 ( σ η) ρη 2 ρ ( ) 1 (ρu) (ρu)( ) 1 (ρu) ρ( ) 1 i j (ρu i u j ). t (3.8) Since ρ ( ) 1 (ρu) = ρ( ) 1 (ρu) ρ t ( ) 1 (ρu) t t = ρ( ) 1 (ρu) ρu ( ) 1 (ρu) t = ρ( ) 1 (ρu) ρu( ) 1 (ρu) ρu ( ) 1 (ρu), t we rewrite (3.8) as follows. 2 (ρ log ρ) t (uρ log ρ) ρ γ1 = ρη 2 ρ ( ) 1 ( σ η) ρ( ) 1 (ρu) t ρu( ) 1 (ρu) ρ ( ) 1 i j (ρu i u j ) u ( ) 1 (ρu). (3.9) 12

13 Suppose that (3.9) also hols for {ρ n, u n, f n, η n, σ n } n 1. 2 (ρ n log ρ n ) t (u n ρ n log ρ n ) (ρ n ) γ1 = ρ n (η n ) 2 ρ n ( ) 1 ( σ n η n ) ρ n ( ) 1 (ρ n u n ) t ρ n u n ( ) 1 (ρ n u n ) ρ n ( ) 1 i j (ρ n u n i u n j ) u n ( ) 1 (ρ n u n ). Let s be a weak limit of ρ n log ρ n. By taking the limit of (3.1) for n, 2 s t (us) ρ γ1 = ρη 2 ρ ( ) 1 ( σ η) ρ( ) 1 (ρu) t ρu( ) 1 (ρu) ρ ( ) 1 i j (ρu i u j ) u ( ) 1 (ρu), (3.1) (3.11) where we use Proposition 2.3 (iii). Next, we take the limit to (1.1b). (ρu) t (ρu u) u ( u) ρ γ η 2 = σ η, (3.12) where we use Proposition 2.3 (ii). Let s = ρ log ρ. By following the same calculations above, we have from which we obtain that 2 s t (us) ρρ γ = ρη 2 ρ ( ) 1 ( σ η) ρ( ) 1 (ρu) (3.13) t ρu( ) 1 (ρu) ρ ( ) 1 i j (ρu i u j ) u ( ) 1 (ρu). Comparing (3.11) an (3.13), we have Since (s s) t (u(s s)) 1 ρ 2 γ1 ρρ γ =. (3.14) s s, ρ γ1 ρρ γ a.e., (s s)x, while (s s )x =. t Therefore, s = s almost everywhere, an ρ n converges strongly to ρ in C(, T ; L 1 ()). Remark 2 In Section 4, we show the strong convergence of ρ n in C(, T ; L 1 ()) using (2.1) with appropriate cut-off functions approximating ρ log ρ. 3.5 Proof of Proposition 2.3 (vi) We now prove Proposition 2.3 (vi) assuming that we have alreay resolve Proposition 2.3 (v). We will prove Proposition 2.3 (iv) an (v) in the next section. Let us begin with the convergence 13

14 of ρ n u n. First, we show that (ρ n u n ) ɛ converges to ρ n u n in L 2 (, T ; L 1 (R 3 )). Here, we extene functions to zero outsie, an g ɛ = g k ɛ an k ɛ is the usual mollifier. Since ( (ρ n u n ) ɛ ρ n u n) (x) = ρ n (y, t) ρ n (x, t) u n (y, t)k ɛ (x y)y ρ n (x, t) u n ɛ (x, t) u n (x, t), we have ( (ρ n u n ) ɛ ρ n u n) (x) x sup z ɛ x ρ n (y, t) ρ n (x, t) p k ɛ (x y) ρ n ( z) ρ n L p () Now, we choose p > 6 p 5 such that Moreover, (2.18) implies that sup z ɛ 1 p ( u ) n p p 1 p p 1 ɛ L 1 () ρn L p () u n ɛ u n p L u n L p p 1 () ρn L p () u n ɛ u n L p p 1 (). p 1 () p 1 < 6. Then, un ɛ u n p converges to as ɛ goes to. L p 1 () ρ n ( z) ρ n L p () converges to as ɛ goes to. Therefore, (ρ n u n ) ɛ converges to ρ n u n in L 2 (, T ; L 1 ()) as ɛ goes to, uniformly in n. Next, we euce that (ρ n u n ) ɛ converges to (ρu) ɛ as n goes to. Since (ρ n u n ) ɛ is smooth in x an t (ρn u n ) ɛ is boune in L 2 (, T ; H m (R 3 )) for any m, (ρ n u n ) ɛ converges to (ρu) ɛ as n goes to in L 1 (R 3 (, T )) for each ɛ >. Since ρ n u n is uniformly boune in L (, T ; L 2γ γ1 ()) an ρ n u n ρu L 1 ( (,T )) ρ n u n (ρ n u n ) ɛ L 1 ( (,T )) (ρ n u n ) ɛ (ρu) ɛ L 1 ( (,T )) (ρu) ɛ ρu L 1 ( (,T )), ρ n u n converges to ρu in L 1 ( (, T )). The secon statement in (2.19) is immeiate consequence of the uniform boun of u n in L 2 ( (, T )). Since ρ n u u n 2 = ρ n u n 2 2ρ n u n u ρ n u 2 converges to in L 1 (), the thir statement is prove. Finally, three previous results implies ρ n u n i un j converges to ρu iu j almost everywhere, an ρ n u n i un j is uniformly boune in L (, T ; L 1 ()) L 1 (, T ; L p ()), 1 p = 1 γ 1 3 < 1. Therefore, the last statement in (2.19) hols. 4 Proof of Proposition 2.3 (iv) an (v) The strong convergence of {ρ (n) } n 1 can be prove by introucing a family of cut-off functions approximating ρ log ρ in the renormalize solution setting. First, we show that ρ satisfies higher integrability conition. 4.1 Higher integrability of ρ We first efine the inverse of the ivergence operator. We enote the solution v of v = g in, v = on. 14

15 by v = T g. This operator T = (T 1, T 2, T 3 ) is the inverse of the ivergence operator such that { } T : g L p ; gx = W 1,p (), with the following bouneness property: T (g) W 1,p () C g L p (). If in aition g can be written as g = h for a certain h L r with h ˆn = on, then T (g) L r () C h L r (). We will use this operator to obtain higher integrability of ρ. By extening (2.1) to zero outsie an regularizing it, we have, ( b t b(ρ) ɛ (b(ρ) ɛ u) (ρ)ρ b(ρ) ) u = r ɛ, (4.1) ɛ where b(ρ) ɛ = b(ρ) g ɛ. As prove in 13, we have We are now reay to prove the following result. r ɛ in L 2 (R 3 (, T )). (4.2) Lemma 4.1 Let γ > 3 2 an {ρ, u, f, η, σ} be a weak solution of the system (1.1). Then, there exists θ >, epening only γ, such that Proof: We take a test function of the form φ i = χ(t)t i b(ρ) ɛ b(ρ) ɛ y, ρ L γθ ( (,T )) C(T ). an test it against (1.1b). Then, with the ai of (4.1), T χρ γ b(ρ) ɛ xt T = T T T T T χρ γ T b(ρ) ɛ y xt χ t ρu T ((b χρu T (ρ)ρ b(ρ)) u ) ɛ = I 1 I 11. T χρu T r ɛ r ɛ y xt χρu T T χρu i u j i T j b(ρ) ɛ b(ρ) ɛ y xt χ u b(ρ) ɛ χσ ij i T j b(ρ) ɛ b(ρ) ɛ y xt b(ρ) ɛ y = 1 b(ρ) ɛ y, χ D(, T ) T b(ρ) ɛ y xt b(ρ) ɛ b(ρ) ɛ y xt ( (b (ρ)ρ b(ρ)) u ) ɛ y xt T 15 (b(ρ) ɛ u ) xt χ i u j i T j b(ρ) ɛ χη 2 b(ρ) ɛ χη b(ρ) ɛ b(ρ) ɛ y xt b(ρ) ɛ y xt b(ρ) ɛ y xt

16 We now estimate I 1,, I 11. For etails, see 1. I 1 C(T ). I 2 ρu L (,T );L γ1 2γ b(ρ) ɛ () L (,T ;L 5γ 3 6γ C(T ) b(ρ) ɛ ()) L (,T ;L 5γ 3 6γ. ()) I 3 ρ L (,T ;L γ ()) u 2 L 2 ( (,T )) b(ρ) ɛ L (,T ;L 2γ 3 3γ C(T ) b(ρ) ɛ ()) L (,T ;L 2γ 3 3γ. ()) I 4 ρu L (,T ;L γ1 2γ r ɛ L ()) 2 ( (,T )) C(T ) r ɛ L 2 ( (,T )). I 5 I 6 ρ L (,T ;L γ ()) u 2 L 2 ( (,T )) b(ρ) ɛ L (,T ;L 2γ 3 3γ C(T ) b(ρ) ɛ ()) L (,T ;L 2γ 3 3γ. ()) I 7 I 8 u L 2 ( (,T )) b(ρ) ɛ L 2 ( (,T )) C(T ) b(ρ) ɛ L 2 ( (,T )). ( ) I 9 I 1 I 11 η 2 L 2 (,T ;L 6 ()) σ L 1 (,T ;L 3 ()) η L 1 (,T ;L 3 ()) b(ρ) ɛ L (,T ;L 2 3 ()) C(T ) b(ρ) ɛ L (,T ;L 2 3. ()) In sum, T χρ γ (b(ρ)) ɛ xt C(T ) b(ρ) ɛ L (,T ;L 5γ 3 6γ b(ρ) ɛ ()) L (,T ;L 2γ 3 3γ ()) b(ρ) ɛ L (,T ;L 3 2 ()) b(ρ) ɛ L 2 ( (,T )) r ɛ L 2 ( (,T )). By taking the limit in ɛ, T χρ γ b(ρ)xt C(T ) b(ρ) L (,T ;L 5γ 3 6γ ()) b(ρ) L (,T ;L 2γ 3 3γ ()) b(ρ) L (,T ;L 3 2 ()) b(ρ) L 2 ( (,T )). (4.3) (4.4) We approximate z z θ by a sequence of {b n } in (2.1), an approximate χ to the ientity function of (, T ). Then, We note that 3 2, for γ < 6, 3γ 2γ 3 T ρ γθ xt C(T ) ρ θ L (,T ;L 5γ 3 6γ ρ θ ()) L (,T ;L 2γ 3 3γ ()) ρ θ L (,T ;L 3 2 ()) ρθ L 2 ( (,T )). 6γ 5γ 3 < 3γ 2γ 3. The relation between 3γ 2 for γ 6. In either cases, we take θ such that 3γ 2γ 3 an 2 epens on the range of γ: (4.5) 2γ 3 > 2 3γ θ γ. (4.6) 2γ 3 Then, which completes the proof. T ρ γθ xt C(T ) (4.7) 16

17 Remark 3 From (4.6), the best possible θ is 2 3γ 1, an higher integrability of ρ can be obtaine by choosing appropriate cut-off functions in (2.1) in the spirit of Feireisl 1. If γ 9 5, then γ θ 2. In 13, Lions use this iea in orer to show higher integrability of ρ by multiplying (3.7) by ρ θ. 4.2 Limit of the effective viscous flux We now stuy the limit of the so calle the effective viscous flux, ρ γ 2 u. In the formal proof, we take ( ) 1 to (1.1b) an multiply by ρ. In this section, we instea take χ(t)φ(x) 1 T k (ρ n ) as test functions to (2.1) to obtain a better convergence result on the effective viscous flux. Here, the cut-off function T k is efine as where T C (R) satisfies T k (z) = kt ( z ), (4.8) k T (z) = z for z 1, T (z) = 2 for z 3, T is concave on, ), an T ( z) = T (z). From (2.1), t T k (ρ n ) (T k (ρ n )u n ) ( T k (ρn )ρ n T k (ρ n ) ) u n = (4.9) hols in the sense of the istributions. Passing to the limit in (4.9), we have t T k (ρ) (T k (ρ)u) (T k (ρ)ρ T k(ρ)) u = (4.1) in the sense of istributions. To take the limit to the effective viscous flux, we nee the following lemma. For the proof, we refer the reaer to Feireisl 1. Lemma 4.2 Suppose v n v in L p (), w n w in L q (), with 1 p 1 q = 1 r < 1. Then, where R ij = i j 1. v n R ij (w n ) w n R ij (v n ) vr ij (w) wr ij (v) in L r (), Lemma 4.3 Uner the conition in Proposition 2.3, we have T lim χφ (ρ n ) γ 2 u n T T k (ρ n )xt = χφ ρ γ 2 u T k (ρ)xt n for χ D(, T ) an φ D(). 17

18 Proof: We take χ(t)φ(x) 1 T k (ρ n ) as test functions to (1.1b). Then, T T = T T T T T T T χφ (ρ n ) γ 2 u n T k (ρ n )xt χ u n (ρ n ) γ φ 1 T k (ρ n )xt χ φ u n 1 T k (ρ n ) u n i j φ j 1 i T k (ρ n ) xt T χu n φt k (ρ n )xt χρ n u n i u n j j φ 1 i T k (ρ n )xt φρ n u n χ t 1 T k (ρ n ) χ 1 ( T k (ρ n ) T k (ρn )ρ n) u n xt χφu n i T k (ρ n )R ij (ρ n u n j ) ρ n u n j R ij T k (ρ n ) xt T χ(η n ) 2 φ 1 T k (ρ n )xt χφ(η n ) 2 T k (ρ n )xt T χσ ij i φ 1 j T k (ρ n )xt χη φ 1 T k (ρ n )xt We now take the limit of (4.11) for n. T T = T T T T T T T T χφ ρ γ 2 u T k (ρ)xt χφσ ij i 1 j T k (ρ n )xt χφηt k (ρ n )xt χ u ρ γ φ 1 T k (ρ)xt χ φ u 1 T k (ρ) u i j φ j 1 i T k (ρ) xt T χu φt k (ρ)xt χρu i u j j φ 1 i T k (ρ)xt φρu χ t 1 T k (ρ) χ 1 ( T k (ρ) T k (ρ)ρ) u xt χφu i T k (ρ)r ij (ρu j ) ρ n u j R ij T k (ρ) xt. T χη 2 φ 1 T k (ρ)xt χσ ij i φ 1 j T k (ρ)xt χη φ 1 T k (ρ)xt T T χφη 2 T k (ρ)xt χφσ ij i 1 j T k (ρ)xt χφηt k (ρ)xt, (4.11) (4.12) 18

19 where we use lemma 4.2 to show T lim χφu n n i T k (ρ n )R ij (ρ n u n j ) ρ n u n j R ij T k (ρ n ) xt T = χφu i T k (ρ)r i,j (ρu j ) ρ n u j R i,j T k (ρ) xt. (4.13) We now take χφ 1 T k (ρ) as test functions to (ρu) t (ρu u) u ( u) ρ γ η 2 = σ η an o the same calculation using (4.1). Then, T T = T T T T T T T χφ ρ γ 2 u T k (ρ)xt χ u ρ γ φ 1 T k (ρ)xt χ φ u 1 T k (ρ) u i j φ j 1 i T k (ρ) xt T χu φt k (ρ)xt χρu i u j j φ 1 i T k (ρ)xt φρu χ t 1 T k (ρ) χ 1 ( T k (ρ) T k (ρ)ρ) u xt χφu i T k (ρ)r ij (ρu j ) ρ n u j R ij T k (ρ) xt T χη 2 φ 1 T k (ρ)xt χσ ij i φ 1 j T k (ρ)xt χη φ 1 T k (ρ)xt T T χφη 2 T k (ρ)xt χφσ ij i 1 j T k (ρ)xt χφηt k (ρ)xt. (4.14) Therefore, by comparing (4.12) an (4.14), we complete the proof. Corollary 4.4 Let ρ be a weak limit of the sequence {ρ n }. Then, lim sup T k (ρ n ) T k (ρ) L C(T ). (4.15) n γ1 ( (,T )) Proof: As z z γ is convex, T k is concave, an (z γ y γ )(T k (z) T k (y)) T k (z) T k (y) γ1, 19

20 we have lim sup n lim T T n T lim n T = lim n T k (ρ n ) T k (ρ) γ1 xt ((ρ n ) γ ρ γ )(T k (ρ n )T k (ρ))xt T ((ρ n ) γ ρ γ )(T k (ρ n ) T k (ρ))xt (ρ γ ρ γ )(T k (ρ) T k (ρ))xt (ρ n ) γ T k (ρ n ) ρ γ T k (ρ)xt By lemma 4.3, the last term in (4.16) can be estimate as follows. T lim n = lim T n T = lim n 2 sup n (ρ n ) γ T k (ρ n ) ρ γ T k (ρ)xt ( u n )T k (ρ n ) ( u)t k (ρ)xt T k (ρ n ) T k (ρ) T k (ρ) T k (ρ) ( u n )xx u n L 2 ( (,T )) lim sup n Since γ 1 > 2, (4.16) an (4.17) implies (4.15). T k (ρ n ) T k (ρ) L 2 ( (,T )). (4.16) (4.17) 4.3 Proof of Proposition 2.3. (iv) To show that ρ an u solve (1.1a) in the sense of renormalize solutions, we regularize (4.1) to get (Tk t T k (ρ) (ρ)u ) (T ɛ ɛ k (ρ)ρ T k(ρ)) u = r ɛ, (4.18) ɛ where r ɛ in L 2 (R 3 (, T )). We multiply (4.18) by b ( ) T k (ρ) an take ɛ. Then, ( ) ( ) t b T k (ρ) b T k (ρ) u b ( ) ( ) T k (ρ) T k (ρ) b T k (ρ) ( u) = b ( T k (ρ)) (Tk (ρ) T k (ρ)ρ) u in the sense of istributions. By the interpolation, ɛ (4.19) T k (ρ n ) T k (ρn )ρ n L 2 ( (,T )) T k (ρ n ) T k (ρn )ρ n α L 1 ( (,T )) T k(ρ n ) T k (ρn )ρ n 1 α L γ1 ( (,T )), where α = γ 1 γ. Since T k (ρ n ) T k (ρn )ρ n L 1 ( (,T )) k 1 γ sup ρ n γ L n γ ( (,T )) an lim sup T k (ρ n ) T k (ρn )ρ n L γ1 ( (,T )) C(T ) n 2

21 from corollary 4.4, we have b ( T k (ρ)) (Tk (ρ) T k (ρ)ρ) u in L 1 ( (, T )) as k. This completes Proposition 2.3 (iv). 4.4 Proof of Proposition 2.3. (v) We introuce a family of functions L k such that z log z for z < k, L k (z) = z T k (s) z log k z s 2 s for z k. Setting we have On the other han, by (1.1a), L k (z) = β k (z) b k (z), b k (z) C(k), b k (z)z b k(z) = T k (z), By (4.2) an (4.21), for φ Cc () Lk (ρ n ) L k (ρ) (t)φx k t L k (ρ n ) (L k (ρ n )u n ) T k (ρ n ) u n =. (4.2) t L k (ρ) (L k (ρ)u) T k (ρ) u =. (4.21) = t Lk (ρ n ) L k (ρ ) φx Passing to the limit in (4.22) for n, Lk (ρ) L k (ρ) (t)φx t = Lk (ρ n )u n L k (ρ)u φ T k (ρ) u T k (ρ n ) u n φxs Lk (ρ)u L k (ρ)u φxs lim n By approximating φ to the ientity function of, Lk (ρ) L k (ρ) (t)x t Tk (ρ) u T k (ρ n ) u n φxs (4.22) (4.23) t = t t T k (ρ)( u)xt lim n Tk (ρ) T k (ρ) ( u)xs. T k (ρ n )( u n )xs (4.24) We approximate z log z by L k (z). By corollary 4.4, with γ 1 > 2, the right-han sie of (4.24) tens to as k. Therefore, ρ log ρ(t) = (ρ log ρ)(t), for all t, T (4.25) which implies the strong convergence of {ρ n } in L 1 ( (, T )). Proposition 2.3 (v). This completes the proof of 21

22 5 Construction of approximate sequences We now construct an approximate sequence of solutions to the system (1.1) so that we can apply the compactness argument (Proposition 2.3) to obtain a weak solution. This section consists of three parts: the regularization of ρ t ρu in (1.1b), nonlinear amping to the equation of ρ an η, an the truncation of ρ γ an η 2. Collecting all these steps, we can prove Theorem Smoothing ρ t ρu We consier the following system of equations: ρ t (ρu) =, (5.1a) (ρ ɛ u) t ((ρu) ɛ u) u ( u) ρ γ η 2 = σ ɛ η ɛ, (5.1b) f t (u ɛ f) τ (P τ ( x u ɛ τ)f) τ f f =. (5.1c) In Section 5.2 will be shown that there exists a solution {ρ, u, f, η, σ} to (5.1) such that ρ L p ( (, T )), p = 5 3γ. By the energy ientity t ρɛ u 2 ργ 2 γ 1 η2 ψ x 4 τ f 2 τx 4 f 2 τx S 2 u 2 u 2 2 η 2 x =, we can obtain the following a priori estimates: (5.2) ρ C(, T ; L 1 ()) L (, T ; L γ ()), ρ ɛ u 2 L (, T ; L 1 ()), u L 2 (, T ; H 1 ()) ρ ɛ u L (, T ; L 2γ γ1 ()) L 2 (, T ; L r 1 ()), r = γ f L 2 (, T ; L 6 5 ( )), σ L 1 (, T ; L 3 ()) L (, T ; L 2 ()), η L (, T ; L 2 ()) L 2 (, T ; H 1 ()), σ L 1 (, T ; L 3 2 ()) L 2 (, T ; L 4 3 ()). (5.3) If u L 2 (, T ; H 1 ()), (5.1c) has a smooth solution, an the entropy ψ an its equation make sense. Thus, we only focus on the existence of a solution {ρ, u, η} uner the assumption that f is smooth. By taking θ = 2 3γ 1 in Lemma 4.1, we have ρ L p ( (, T )), p = 5 γ 1. (5.4) Nonlinear amping We wish to buil solutions of (5.1) as a limit of the following system of equations: ρ t (ρu) δρ q =, (5.5a) (ρ ɛ u) t ((ρu) ɛ u) u ( u) ρ γ η 2 δ(ρ q ) ɛ (η m ) ɛ u = σ ɛ η ɛ, (5.5b) f t (u ɛ f) τ (P τ ( x u ɛ τ)f) τ f f =, (5.5c) η t (ηu) η δη m =. (5.5) 22

23 where q > γ 1 an m > 3, with m q, will be etermine later. We note that we a a nonlinear amping to the equation of η, not the equation of f. We will prove the existence of a solution {ρ, u, f, η, σ} to (5.5) satisfying u L s (, T ; W 1,s ()), u t L s (, T ; W 1,s ()), ρ, η C(, T ; L s ()), 1 s < ρ ɛ C 1 (, T ; C k ( )), (ρu) ɛ C(, T ; C k ( )), k in Section 5.3. From (5.5), we have the following energy ientity: ρɛ u 2 ργ t 2 γ 1 η2 ψ x 4 τ f 2 τx 4 f 2 τx S 2 T T T δ (ρ q ) ɛ u 2 xt δ ρ qγ 1 xt δ (η m ) ɛ u 2 xt δ T Therefore, we have η m1 xt u 2 u 2 2 η 2 x =. ρ C(, T ; L 1 ()) L (, T ; L γ ()), ρ ɛ u 2 L (, T ; L 1 ()), u L 2 (, T ; H 1 ()) T T δ (ρ q ) ɛ u 2 xt C, δ ρ qγ 1 xt C, ρ ɛ u L (, T ; L 2γ γ1 ()) L 2 (, T ; L r ()), 1 r = γ f L 2 (, T ; L 6 5 ( )), η L (, T ; L 2 ()) L 2 (, T ; H 1 ()), T T δ (η m ) ɛ u 2 xt C, δ η m1 xt C, σ L 1 (, T ; L 3 ()) L (, T ; L 2 ()), σ L 1 (, T ; L 3 2 ()) L 2 (, T ; L 4 3 ()). (5.6) (5.7) (5.8) Moreover, for a sufficiently small δ, hols. To prove this, we integrate (5.5a) over. t ρ(t)x = ρ,ɛ x δ ρ q (s)xs ρ x δ ρ ɛ 1 C on, T (5.9) T qα T 1 ρ x δt qγ 1 ρ x C(δT ) qα qγ 1 δt ρ qα I {ρ>1} xt δ T ρ x δ T ρ qγ 1 xt ρ x where < α < γ 1. By taking a sufficiently small δ, we have ρ(t)x 1 ρ x 2 23 qα qγ 1 ρ q (t)xt ρi {ρ<1} xt δt ρ x

24 from which we have ρ ɛ (x, t) = ρ(y, t)k ɛ (x y)y inf k ɛ (z) ρ(y)y 1 z 2 C. (5.1) Since ρ ɛ u 2 L (, T ; L 1 ()), u L (, T ; L 2 ()). Moreover, from ρ L (, T ; L 1 ()), ρ ɛ L ( (, T )). Next, we obtain higher integrability of ρ by using (3.9), with replacing the multiplication of ρ by ρ θ to (3.8). We also have aitional terms from the amping terms. ρ γθ = t ρ θ ( ) 1 (ρ ɛ u) uρ θ ( ) 1 (ρ ɛ u) 2( u)ρ θ (θ 1)( u)ρ θ ( ) 1 (ρ ɛ u) ρ θ ( ) 1 ( σ ɛ η ɛ ) ρ θ R i R j ((ρu i ) ɛ u j ) u j R i R j (ρ ɛ u i ) ρ θ η 2 (5.11) δρ θ ( ) 1 ((ρ q ) ɛ u) θδρ qθ 1 ( ) 1 (ρ ɛ u) δρ θ ( ) 1 ((η m ) ɛ u). Integrating (5.11) over (, T ), we have T T ρ γθ xt C(T ) T δ δ δ T T T T ( u)ρ θ T xt ρ θ ( ) 1 ( σ ɛ η ɛ ) xt ( u)ρ θ ( ) 1 (ρ ɛ u) xt T ρ θ xt R i R j ((ρu i ) ɛ u j ) u j R i R j (ρ ɛ u i ) ρ θ ( ) 1 ((ρ q ) ɛ u) xt ρ θ ( ) 1 ((η m ) ɛ u) xt ρ qθ 1 ( ) 1 (ρ ɛ u) xt. ρ θ η 2 xt (5.12) Let θ = 2 3γ. We only provie estimations involving δ. The rest of them are easily by the regularization. First, δ ρ θ ( ) 1 ((ρ q ) ɛ u) L 1 ( (,T )) δ ρ θ L (,T ;L γ ()) (ρ q ) ɛ 2(qγ 1) L q (,T ;L ()) ρ ɛ u L 2 (,T ;L 2 ()) δ ρ θ L (,T ;L γ ()) ɛ α ρ q 2(qγ 1) L q ( (,T )) ρ ɛ u L 2 (,T ;L 2 ()) ɛ α δδ 2 2(qγ 1) δ 1 2 as δ, (5.13) where < 1 2 q 2(qγ 1) < 1. Similarly, δ ρ θ ( ) 1 ((η m ) ɛ u) L 1 ( (,T )) ɛ β δδ 1 m δ 1 2 as δ, (5.14) 24

25 where < m < 1. Next, we want to estimate δρpθ 1 ( ) 1 (ρ ɛ u). Since ρ ɛ u L (, T ; L 2 ()) L 2 (, T ; L 6 ()) an we have δ ρ qθ 1 qγ 1 L qθ 1 ( (,T )) δδ qθ 1 qγ 1, δ ρ qθ 1 ( ) 1 (ρ ɛ u) L 1 ( (,T )) as δ (5.15) by taking q > γ 1, which is close to γ 1. Therefore, as δ, ρ δ is boune in L (, T ; L γ ()) L p ( (, T )), p = 5 3 γ u δ is boune in L (, T ; L 2 ()) L 2 (, T ; H 1 ()) (5.16) so we can pass to the limit to (5.5) to obtain a solution of (5.1) by the compactness argument in Section 4. Inee, (ρ δ ) ɛ ρ ɛ L r ( (, T )), u δ u L r (, T ; L 2 ()) 1 r < so the analysis is easier than the case without ɛ. 5.3 Truncation of the pressure In this section, we construct a solution of (5.5) by ρ t (ρu) δρ q =, (5.17a) (ρ ɛ u) t ((ρu) ɛ u) u ( u) T (R) δ(ρ q ) ɛ (η m ) ɛ u = σ ɛ η ɛ, (5.17b) f t (u ɛ f) τ (P τ ( x u ɛ τ)f) τ f f =, η t (ηu) η δη m =. (5.17c) (5.17) where T (R) = (ρ R) γ (η R) 2 an f R = min{f, R}. We will show at the en of this section that there exists a smooth solution to (5.17) such that ρ, η C(, T ; W 1,s ()), u L s (, T ; W 2,s ()), u t L s ( (, T )), 1 s <. (5.18) We begin with the energy ientity of (5.17). ρɛ u 2 A R (ρ) B R (η) ψ η x 4 τ f 2 τx t 2 S 2 4 T T f 2 τx δ (ρ q ) ɛ u 2 xt δ ρ q A R(ρ)xt T T δ (η m ) ɛ u 2 xt δ ρ m B R(η)xt where A R (ρ) = ρ ρ u 2 u 2 2 (η R) 2 x, (t R) γ t, A R(ρ) = t 2 γ(ρ η R)γ 1, B R (η) = η γ 1 25 (5.19) (t R) 2 t, B R(η) = 2(η R). t 2

26 From (5.19), we have ρ C(, T ; L 1 ()), ρ ɛ u 2 L (, T ; L 1 ()), u L 2 (, T ; H 1 ()) A R (ρ) L (, T ; L 1 ()), A R(ρ)ρ q L 1 ( (, T )), T T δ (ρ q ) ɛ u 2 xt C, δ ρ qγ 1 xt C, ρ ɛ u L (, T ; L 2γ γ1 ()) L 2 (, T ; L r ()), 1 r = γ f L 2 (, T ; L 6 5 ( )), B R (η) L (, T ; L 2 ()), η R L 2 (, T ; H 1 ()), σ L 1 (, T ; L 3 ()) L (, T ; L 2 ()), σ L 1 (, T ; L 3 2 ()) L 2 (, T ; L 4 3 ()). (5.2) We still have ρ ɛ 1 C uniformly in R 1 as for sufficiently small δ so that u L (, T ; L 2 ()) an (ρu) ɛ L p (, T ; C k ( )). We are going to prove ρ L (, T ; L s ()), u L s (, t; W 1,s ()), u t L s (, T ; W 1,s ()), 1 < s <. (5.21) For any 1 < r <, By taking s = qr 1 q 1, t ρr (uρ r ) δrρ qr 1 = (r 1)( u)ρ r. ρ r s L (,T ;L r ()) ρq 1 L s ( (,T )) u L s ( (,T )). (5.22) We can estimate η by using the same metho. We set a = mr 1 m 1. For m q, s > a. Therefore, η r a L (,T ;L r ()) ηm 1 L a ( (,T )) u L s ( (,T )). (5.23) We now estimate u by using the estimation of the heat equation (13). By writing (ρu) ɛ u = ((ρu) ɛ u) ( (ρu) ɛ )u, u t L s (,T ;W 1,s ()) u L s (,T ;W 1,s ()) C(T ) (ρ R) γ L s ( (,T )) (ρu) ɛ u L s ( (,T )) ( (ρu) ɛ )u L s ( (,T )) σ ɛ L s ( (,T )) η ɛ L s ( (,T )) η 2 L 2s ( (,T )) (5.24) C(T ) ρ γ L γs ( (,T )) ρu L ls l s (,T ;L 1 ()) u L l (,T ;L s ()) η 2 L 2s ( (,T )) for any s l. Since q > γ1 an m > 3, (m 1)a > 2s an (q 1)s > γs. Therefore, we can obtain (5.21) by (5.22), (5.23), an (5.24) provie that we can boun ρu L ls l s (,T ;L 1 ()) u L l (,T ;L s ()) in (5.24). This can be one by a bootstrap argument. First, we take s = 2, l =. Then, ρ L (, T ; L q 1 ()), thus ρ L (, T ; L γ ()). (5.25) If γ 2, then ρu L (, T ; L 1 ()) so that u t L s (,T ;W 1,s ()) u L s (,T ;W 1,s ()) C(T ) ρ γ L γs ( (,T )) u L s ( (,T )) η 2 L 2s ( (,T )). (5.26) 26

27 We note that we o not have σ ɛ η ɛ in the right-han sie of (5.26) ue to the regularization. If γ < 2, then we take s = 1q 3q4, k = 1q ks 3q 6 so that q = k s. Then, ρ L (, T ; L w ()), w = s 1 q 1. (5.27) Since w > 2, ρu L (, T ; L 1 ()). Therefore, we can conclue as the previous case. Now, we want to pass to the limit in R. Since T ρ θ (ρ R) γ ρ θγ L 1 ( (,T )) ρ θγ I {ρ>r} 1 T ρ θγ 1 xt 1 R R, we have ρ θ (ρ R) γ ρ θγ L 1 ( (,T )). Similarly, ρ θ (η R) 2 ρ θ η 2 L 1 ( (,T )). Therefore, we can recover a solution of (5.5) satisfying (5.6) in Section 5.2. It remains to show (5.18). First, we show that ρ L ( (, T )). From the equation of ρ, t (log ρ) u log ρ u δρp 1 = (5.28) Moreover, u = 1 2 (ρ R)γ 1 1 R) 2 (ρ γ x 1 2 (η R)2 1 1 (η R) 2 x 2 ( ) 1 ( σ ɛ η ɛ ) ( ) 1 (δ(ρ q ) ɛ u) ( ) 1 (δ(η m ) ɛ u) (5.29) R i R j ((ρu i ) ɛ u j ) t ( ) 1 (ρ ɛ u) Let Φ = ( ) 1 (ρ ɛ u). By (5.28) an (5.29), t (log ρ Φ) u (log ρ Φ) δρp 1 = Ψ L ( (, T )). (5.3) By the maximum principle, log ρ Φ C. Therefore, ρ L ( (, T )). Next, we estimate ρ. For any 1 r <, which implies that Similarly, t ρ r (u ρ r ) u ρ r 2 u ρ r 1 (5.31) t ρ L r () u L () ρ L r () 2 u L r (). (5.32) t η L r () u L () η L r () 2 u L r (). (5.33) We nee to estimate erivatives of u. From stanar parabolic estimates, 2 u L r ( (,T )) C(T ) (ρ R) γ L r ( (,T )) (η R) 2 L r ( (,T )) By taking r > 3, u(t) L () log C(T ) ρ L r ( (,T )) η L r ( (,T )). (5.34) ( ) C(T ) max ρ(τ) L τ t r () η(τ) L r () L () q. (5.35) By the Gronwall s inequality, ρ L (,T ;L r ()) η L (,T ;L r ()) C(T ), which implies that u L ( (, T )) an 2 u L r ( (, T )). Therefore, from the equation of u, u t L r ( (, T )). This completes the proof. 27

28 6 Appenix 6.1 Appenix 1: erivation of the equation of ψ We take the time erivative to ψ = f ln fτ. Then, ψ t = f t ln f f t τ = (I) (II). We obtain (I) an (II) by using the equation of f (1.1c). (I) = (uf) τ (P τ ( x uτ)f) τ f f ln fτ = (a) (b) (c) (). (a) = (uf) ln fτ = (uf ln f)τ u x fτ = (uψ) u fτ. (b) = τ (P τ ( x uτ)f) ln fτ = uτf τ f f (6.1) = u : τ τ fτ = u : σ. (6.2) The proof of last equality in relation (6.2) requires the expression of these quantities in spherical coorinates. We refer the reaer to 19 for further etails. (c) = τ ffτ = τ (f ln f) 2 τ f τ ln f f (ln f) τ S2 τ f 2 = 2 f τ τ f f f τ S2 τ f 2 S2 τ f 2 = 2 τ τ fτ τ f f = 4 τ f 2 τ. () = x ffτ = x (f ln f) 2 x f x ln f f x (ln f) τ S2 x f 2 S2 x f 2 = x ψ 2 τ x f τ f f = x ψ 4 τ f 2 τ x η. Now, we calculate (II). (II) = (uf) τ (P τ ( x uτ)f) τ f x f τ = (uη) x η. (6.5) (6.3) (6.4) 28

29 Collecting all terms in (6.1) - (6.5), ψ t = (uψ) u η u : σ 4 τ f 2 τ x ψ 4 x f 2 τ η (uη) η = (uψ) ψ 4 τ f 2 τ 4 x f 2 τ u : σ ( u)η. (6.6) 6.2 Appenix 2: verification of the formal proof in Section 3.4 In this section, we provie a rigorous proof of the formal proof in Section 3.4. First, we verify (3.6). From the equation of the ensity, ρ t (ρu) =, we can obtain t β(ρ) (uβ(ρ)) ( u) ρβ (ρ) β(ρ) =, (6.7) where β is a C 1 function from, to R such that β (t) C(1 t α ), α = q 2 2. We note that (6.7) makes sense because u L 2 (, T ; H 1 loc ()). We approximate ρ log ρ by β δ(ρ) = ρ log(ρ δ). Then, ρβ δ (ρ) β δ(ρ) in L q, which verifies (3.6) in the sense of istributions. Next, we want to verify (3.8) by showing that each term in the right-han sie of (3.7) can be multiplie by ρ to be in L 1 ( (, T )). ρ ρ γ is from ρ L γ1 (). Since ρ L (, T ; L s ), s > 3 an η 2 L 1 (, T ; L 3 ), the prouct ρη 2 is in L 1 ( (, T )). ρ( ) 1 ( σ η) ρ(σ ηi) L (, T ; L s ()) L 1 (, T ; L 3 ()). Since s > 3, 1 s ρ( ) 1 i j (ρu i u j ) ρρu u: Since u u L 1 (, T ; L 3 ()), ρu u L 1 (, T ; L r ()) L p (, T ; L q ()), 1 r = 1 s 1 3 1, 1 q = 1 rp 1 1 p. Since ρ L (, T ; L s ), ρρu u L 1 ( (, T )), with 1 s 1 r = 2 s = 1. We rewrite ρ t ( ) 1 (ρu) as ρ ( ) 1 (ρu) = ρ( ) 1 (ρu) ρ t ( ) 1 (ρu) t t = ρ( ) 1 (ρu) (ρu)( ) 1 (ρu) t = ρ( ) 1 (ρu) (ρu)( ) 1 (ρu) (ρu) ( ) 1 (ρu). t First, (ρu) ( ) 1 (ρu) ρρu u which is alreay one. Next, we estimate ρ( ) 1 (ρu). Since ρu L (, T ; L 2γ γ1 ()), ( ) 1 (ρu) L (, T ; L 6γ γ3 ()). Therefore, ρ( ) 1 (ρu) 29

30 L (, T ; L 1 ()) because 1 s γ3 6γ < 1 3 γ3 6γ < 3γ3 6γ < 1. Finally, (ρu)( ) 1 (ρu) makes sense because ( ) 1 (ρu) L (, T ; L 6γ γ3 ()), ρ L (, T ; L s ()), an u L 2 (, T ; L 6 ()), with 1 s 1 6 γ3 6γ < γ3 6γ < 1. Finally, we verify (3.11) by passing to the limit in (3.1). ρ n log ρ n u n : We take g n = ρ n log ρ n an h n = u n. To verify the conition of g n, we can use (4.6). ρ n ( ) 1 ( σ n η n ): We take g n = ρ n, h n = ( ) 1 ( σ n η n ). This is possible because σ n, η n L 1 (, T ; L 3 ()) an ρ n L (, T ; L s ()), s > 3. We alreay show that ρ n (η n ) 2 converges to ρη 2 in the sense of istributions. ρ n ( ) 1 (ρ n u n ) an ρ n u n ( ) 1 (ρ n u n ): Since we have ρ n u n L (, T ; L q ()) L 2 (, T ; L r ()), 1 q = 1 2s 1 2, 1 r = s 1 2. ( ) 1 (ρ n u n ) L (, T ; W 1,q loc ()) L2 (, T ; W 1,r loc ()). Therefore, we can take g n = ρ n, ρ n u n an h n = ( ) 1 (ρ n u n ). ρ ( ) n 1 i j (ρ n u n i un j ) un ( ) 1 (ρ n u n ) = ρ n u n j, R ir j (ρ n u n ): First, we control the commutator. Since u n L 2 (, T ; L 2 ()) an ρ n u n L 2 (, T ; L p ()), with 1 p = s 1 2, Let g n = ρ n u n i u n j, R i R j (ρ n u n ) is boune in L 1 (, T ; W 1,q ()), an h n = u n j. Then, 1 q = p. U n = u n j, R i R j (ρ n u n i ) U = u j, R i R j (ρu i ) L 1 (, T ; L q 2 ()), 1 q 2 < 1 1 s 2 3. in the sense of istributions. Now, let g n = ρ n an h n = U n. Then, ρ n U n converges to ρu in the sense of istributions. This completes the proof of the formal argument in Section 3.4. Acknowlegments H.B. gratefully acknowleges the support provie by the Center for Scientific Computation an Mathematical Moeling (CSCAMM) at the University of Marylan where this research was performe. H.B is partially supporte by NSF grants DMS an FRG K.T. acknowleges the support by the National Science Founation uner the awars DMS an DMS References 1 H. Bae an K. Trivisa, On the Doi moel for the suspensions of ro-like molecules: Globalin-time exisence, To appear in Commun. Math. Sci. (211). 2 J. A. Carrillo an T. Gouon. Stability an Asymptotic Analysis of a Flui-Particle Interaction Moel. Comm. Partial Differential Equations, 31: , J. A. Carrillo, T. Gouon, an P. Lafitte, Simulation of flui an particles flows: asymptotic preserving schemes for bubbling an flowing regimes. J. Comput. Phys. 227: , 28. 3

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