Vlasov-Fokker-Planck equations

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ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Hydrodynamic limit for the inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations Yunfei Su, Lei Yao ∗ School of Mathematics, Northwest University, Xi’an 710127, China Received 23 March 2019; revised 13 December 2019; accepted 29 December 2019

Abstract In this paper, we study the hydrodynamic limit for the inhomogeneous incompressible NavierStokes/Vlasov-Fokker-Planck equations in a two or three dimensional bounded domain when the initial density is bounded away from zero. The proof relies on the relative entropy argument to obtain the strong convergence of macroscopic density of the particles n in L∞ (0, T ; L1 ()), which extends the works of Goudon-Jabin-Vasseur [15] and Mellt-Vasseur [26] to inhomogeneous incompressible NavierStokes/Vlasov-Fokker-Planck equations. Precisely, the relative entropy estimates in [15] and [26] give the strong convergence of u and n , ρ  and n , respectively. However, we only obtain the strong convergence of n and u from the relative entropy estimate, and we use another way to obtain the strong convergence of ρ  via the convergence of u . Furthermore, when the initial density may vanish, taking advantage of compactness result LM →→ H −1 of Orlicz spaces in 2D, we obtain the convergence of n in L∞ (0, T ; H −1 ()), which is used to obtain the relative entropy estimate, thus we also show the hydrodynamic limit for 2D inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations when there is initial vacuum. © 2020 Elsevier Inc. All rights reserved. MSC: 35Q30; 35Q70; 35Q84 Keywords: Hydrodynamic limit; Inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations; Relative entropy method

* Corresponding author.

E-mail addresses: [email protected] (Y. Su), [email protected] (L. Yao). https://doi.org/10.1016/j.jde.2019.12.027 0022-0396/© 2020 Elsevier Inc. All rights reserved.

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1. Introduction In this paper, we investigate the hydrodynamic limit of the inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations in a two or three dimensional bounded domain. This kind of system arises in various industrial application, such as sedimentation, atmospheric pollution modeling, waste water treatment, chemical engineering (see [5,12,17,28]), sprays or aerosols (see [27,31,32]) and so on. The evolution of particles is described through its distribution function f (x, v, t) on the phase point (x, v) ∈  × Rd (d = 2, 3), at time t ∈ [0, T ], which satisfies Vlasov-Fokker-Planck equation ∂t f + v · ∇x f + ∇v · (Fd f − ∇v f ) = 0.

(1.1)

On the other hand, the fluid is described by two macroscopic quantities, a scalar valued function ρ(x, t) ≥ 0, representing the density of the fluid, and a the vector field u(x, t), representing the velocity of the fluid at a given time and position. These quantities verifies the following inhomogeneous incompressible Navier-Stokes equation ⎧ ⎨ ∂t ρ + divx (ρu) = 0, ∂t (ρu) + divx (ρu ⊗ u) + ∇x p − x u = Ff , ⎩ divx u = 0.

(1.2)

The interaction of the fluid and particles is modeled by a friction (or drag) force Fd exerted by the fluid onto the particles. This force is assumed to be proportional to the relative velocity of the fluid and the particles, we set Fd = F0 (u(x, t) − v),

(1.3)

where F0 is assumed to be 1. The right hand side of (1.2)2 is the action of the cloud of particles on the fluid, given by  Ff = − Rd

 Fd f dv = −

(u(x, t) − v)f (x, v, t) dv.

(1.4)

Rd

Existence of solutions for the fluid-particle model has been investigated extensively in the past years. Many researchers studied the global existence for fluid-particles system. The earliest result of the global existence and large time behavior of solutions for the Vlasov-Stokes equations in a bounded domain was proved by Hamdache in [19]. Next, Boudin, Desvillettes, Grandmont and Moussa [3] obtained the global weak solutions for the Navier-Stokes-Vlasov equations in three-dimensional periodic domains. After that, global weak solutions are proved in a moving domain [4]. Chae, Kang and Lee [8] showed the global existence of weak solutions in two and three dimensions, as well as the existence and uniqueness result of smooth solutions in two dimensions. Later then, Wang and Yu [30,33] established global existence of weak solutions to the inhomogeneous incompressible Navier-Stokes-Vlasov equations with density-dependence drag force and incompressible Navier-Stokes-Vlasov equations in a bounded domain, respectively. Recently, uniqueness result of the solution to the 2D Vlasov-Navier-Stokes system was

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proved in [18]. Furthermore, Goudon, He, Moussa and Zhang [16] showed the global existence of smooth solutions with small data for Navier-Stokes/Vlasov-Fokker-Planck equations; Carrillo, Duan and Moussa [7] proved the global existence of classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. For the compressible case, Mellet and Vasseur [25] proved the global existence of weak solution to the compressible Navier-Stokes/VlasovFokker-Planck equations in bounded domain of R3 ; Li, Mu and Wang [24] established the global existence and large time behavior of strong solutions close to equilibrium to the compressible Navier-Stokes/Vlasov-Fokker-Planck equations. The mathematical analysis of other related models, such as Euler-Vlasov equations [2], the coupling of the kinetic flocking model and the incompressible Navier-Stokes equations [6] and so on, which have also received much attentions now. In this paper, we are devoted to studying the following fluid-particle model with the Brownian effect: ⎧ ∂t f  + v · ∇x f  + 1 ∇v · ((u − v)f  − ∇v f  ) = 0, ⎪ ⎪ ⎪ ⎨ ∂t ρ  + divx (ρ  u ) = 0,  ⎪ ∂t (ρ  u ) + divx (ρ  u ⊗ u ) + ∇x p  − x u = 1 d (v − u )f  dv, ⎪  R ⎪ ⎩ divx u = 0,

(1.5)

where x ∈  is the spatial variable,  ⊂ Rd (d = 2, 3) is a bounded domain, t ∈ [0, T ] is the time variable, v ∈ Rd is the velocity of the particles, p  is the pressure. The initial data is f  (x, v, 0) = f0 (x, v),

ρ  (x, 0) = ρ0 (x),

u (x, 0) = u0 (x),

(1.6)

and the boundary conditions u (x, t) = 0,

on ∂,

(1.7)

f  (x, v, t) satisfies the reflective boundary condition, i.e., f  (x, v, t) = f  (x, v ∗ , t) for x ∈ ∂, v · n (x) < 0,

(1.8)

where v ∗ = v − 2(v · n (x))

n(x) is the specular velocity, n (x) is the outward unit normal vector to ∂ for all x ∈ ∂. In recent years, hydrodynamic limit of weak solutions to the fluid-particle system has been extensively studied. For the case of one space dimension, Goudon [13] proved the hydrodynamic limit and stratified limit to the Vlasov equation coupled viscous Burgers equation. For the multidimensional case, by using the method of relative entropy, Goudon, Jabin and Vasseur [14,15] established two hydrodynamic limit results to the Vlasov-Fokker-Planck equation coupled to incompressible Navier-Stokes; Mellet and Vasseur [26] showed the hydrodynamic limit of weak solutions to the Vlasov-Fokker-Planck/compressible Navier-Stokes systems; Recently, Choi and Jung [10] extended this result to density-dependent viscosity case. Carrillo, Choi and Karper [6] proved hydrodynamic limit of weak solutions to a kinetic flocking model coupled with the incompressible Navier-Stokes equations.

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There are also some results about the hydrodynamic limits for other kinds of models, Kang and Vasseur [21] considered the Vlasov-type equation under strong local alignment regime; Karper, Mellet and Trivisa [22] considered the kinetic Cucker-Smale flocking model under the regime of diffusive term and strong local alignment; Recently, Figalli and Kang [11] showed the hydrodynamic limit for the kinetic Cucker-Smale flocking model without diffusive term. The main contribution of this paper is to show the convergence behavior of weak solutions (f  , ρ  , u ) to the limit solution (Mn,u , ρ, u) (Mn,u stands for the Maxwellian with density n and velocity u), where (n, ρ, u) satisfies the following system: ⎧ ∂t n + divx (nu) = 0, ⎪ ⎪ ⎪ ⎨ ∂ ρ + div (ρu) = 0, t x ⎪ ((ρ + n)u) + divx ((ρ + n)u ⊗ u) + ∇x P − x u = 0, ∂ ⎪ t ⎪ ⎩ divx u = 0.

(1.9)

Our work is motivated by the result of Goudon, Jabin and Vasseur [15], which is about the hydrodynamic limit of incompressible Navier-Stokes/Vlasov-Fokker-Planck system over Rd (d = 2, 3), it is as follows: ⎧ ∂t f  + v · ∇x f  + 1 ∇v · ((u − v)f  − ∇v f  ) = 0, ⎪ ⎪ ⎪ ⎨ ∂ u + div (u ⊗ u ) + ∇ p  −  u = 1  (v − u )f  dv, t x x x  Rd  ⎪ divx u = 0, ⎪ ⎪ ⎩  f |t=0 = f0 , u |t=0 = u0 ,

(1.10)

its limit problem with ρ  = 1 + n is ⎧  + divx ( ρ u) = 0, ⎪ ⎨ ∂t ρ ρ u) + divx ( ρ u ⊗ u) − x u + ∇x p = 0. ∂t ( ⎪ ⎩ divx u = 0.

(1.11)

Precisely, Goudon, Jabin and Vasseur introduced the relative entropy   H (f , u ) = 



1 H (f |Mn,u ) dvdx + 2

 |u − u|2 dx,



Rd Rd

Rd

here H (f  |Mn,u ) = f  ln( Mfn,u ) + Mn,u − f  , Mn,u = (2π)− 2 n exp( −|v−u| ). They used the rel2   ative entropy method to obtain H (f , u ) → 0, as  → 0, which implied the strong convergence of u , that is, 

d

u → u

in L∞ (0, T ; L2 (Rd )).

Furthermore, f  → Mn,u

in L∞ (0, T ; L1 (Rd × Rd )),

2

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and in L∞ (0, T ; L1 (Rd )),

n → n where

 n (x, t) = 

f  (x, v, t) dv.

Rd

It is also worth to mentioning the result of Mellet and Vasseur [26], which is about the hydrodynamic limit of Vlasov-Fokker-Planck/compressible Navier-Stokes system in three dimensions, it is as follows: ⎧ ⎨ ∂t f  + v · ∇x f  + 1 divv ((u − v)f  − ∇v f  ) = 0, ∂t ρ  + divx (ρ  u ) = 0, (1.12)  ⎩ ∂t (ρ  u ) + divx (ρ  u ⊗ u ) + ∇x (ρ  )γ − μx u = 1 R3 (v − u )f  dv, its limit problem is ⎧ ⎪ ⎨ ∂t n + divx (nu) = 0, ∂t ρ + divx (ρu) = 0, ⎪ ⎩ ∂t ((ρ + n)u) + divx ((ρ + n)u ⊗ u) + ∇x (ρ γ + n) − μx u = 0, they defined the relative entropy H (U  |U ) = (n + ρ  )

|u − u|2 1 + P1 (ρ  |ρ) + P2 (n |n), 2 γ −1

with P1 (ρ  |ρ) = (ρ  )γ − ρ γ − γρ γ −1 (ρ  − ρ), and P2 (n |n) = n log n − n log n − (log n + 1)(n − n) = n log

n + (n − n ). n

Furthermore, they obtained T  H (U  |U ) dxds 0 

T  (n + ρ  )

= 0 

|u − u|2 1 + P1 (ρ  |ρ) + P2 (n |n) dxds 2 γ −1

(1.13)

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√ ≤ C ,

(1.14)

which implies the strong convergence n → n a.e. and L1loc (0, T ; L1 ()) − strong, p

ρ  → ρ a.e. and Lloc (0, T ; Lp ()) − strong, ∀p < γ , and f  → Mn,u =

n 3

(2π) 2

exp(−

|u − v|2 ), a.e. and L1loc (0, T ; L1 ( × R3 )) − strong. 2

In this paper, we use the relative entropy method to study the hydrodynamic limit for the inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations in a two or three dimensional bounded domain when ρ0 ≥ c. In the process of estimating the relative entropy, we have difficulties in estimating the terms K4 and K6 . To solve it, we assume that ρ0 (x) has a positive lower bound, thus ρ  has the√same positive  t lower bound, then K4 and K6 can t be controlled by C 0 H (f  , ρ  , u ) ds + C  and C 0 H (f  , ρ  , u ) ds + C, respectively, see (5.10) and (5.15); They are key to obtain the relative entropy estimate (3.16). On the other hand, combining Lemma 2.2, Csiszár-Kullback-Pinsker inequality and the strong convergence f  → Mn,u in L∞ (0, T ; L1 ( × Rd )) from the relative entropy estimate (3.16), we can obtain the strong convergence of macroscopic density of the particles n in L∞ (0, T ; L1 ()). Then combine with the strong convergence of n and (5.9), we can pass to the limit of the nonlinear terms n u and n u ⊗ u . Furthermore, we can obtain the strong convergence u → u in L2 (0, T ; Lq ()) (1 ≤ q ≤ 6) from the relative entropy estimate (3.16) and (5.16). Finally, comparing with the result of Goudon, Jabin and Vasseur [15], we still need to obtain the strong convergence of ρ  , combining the convergence of u and (5.14), which leads to the strong convergence of ρ  in L∞ (0, T ; Lp ())(1 ≤ p < ∞). Consequently, we can pass to the limit of the nonlinear terms ρ  u and ρ  u ⊗ u . It should be pointed out that ρ0 ≥ c above is only used in the estimates of K4 and K6 . For the case ρ0 ≥ 0, we can obtain the convergence n → n in L∞ (0, T ; H −1 ()) by using a compactness result LM →→ H −1 of Orlicz spaces in 2D, which is conductive to estimate K4 , see (7.11). Besides, K6 can be estimated within a appropriate small time interval (t ≤ T1 ), see (7.12), both of them are key to obtain the relative entropy estimate (3.17). Thus we also obtain the hydrodynamic limit for 2D inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations when there is initial vacuum. The rest of this paper is organized as follows. In section 2, we introduce some useful results. In Section 3, we state the main result of this paper. In Section 4, we give some a priori estimates. Section 5 is devoted to obtain the relative entropy estimate for the proof of Theorem 3.1. In Section 6, by passing to the limit, we complete the proof of Theorem 3.1. Finally, Theorem 3.2 is proved in Section 7. 2. Preliminaries Firstly, we recall some basic facts of the Orlicz spaces, readers can refer to [1,20]:

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• (Definition of the Orlicz spaces) The Orlicz space LA () is the linear hull of the Orlicz class KA (), that is, the smallest vector space (under pointwise addition and scalar multiplication) that contains KA(). And the functional ⎧ ⎫  ⎨ ⎬ |u(x)| uA = uA, = inf k > 0 : A( ) dx ≤ 1 ⎩ ⎭ k 

is a norm on LA (). • (Property 1) The N -functions M and N given by M = M(s) := (1 + s) ln(1 + s) − s and N = N (s) := es − s − 1 are said to be complementary; each is the complement of the other. Besides, an N -functions M is said to satisfy a global 2 condition, if there is a positive constant a such that for every s ≥ 0, the following inequality holds: M(2s) ≤ aM(s); • (Property 2) Let  be a domain in Rd , then the conjugate space (LM ())∗ of LM () is LN (); • (Property 3) The Orlicz space LM () is EN -weakly compact with N being the complementary N -function to M. Next, we state a compactness lemma. d

Lemma 2.1. ([20]). Let U be a bounded domain in Rd , then LM (U ) →→ H − 2 (U ). Let h: R → R be a strictly convex function. The quantity H (x|y) = h(x) − h(y) − h (y)(x − y), can be used as a way to evaluate how far x is from y. Moreover, by the convexity, one yields x H (x|y) =



x z



(h (z) − h (y)) dz = y

h (r) drdz ≥ 0,

y y

and it vanishes if and only if x = y. In order to obtain the convergence behavior of n from f  , we need to introduce the following lemma. Lemma 2.2. ([15]). For i ∈ {1, 2}. Let fi : RN → R+ . We set n(x, t) = we have  H (n1 |n2 ) ≤ H (f1 |f2 ) dv. Rd

 Rd

f (x, v, t) dv. Then,

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Lemma 2.3. ([29]). Assume X ⊂ E ⊂ Y are Banach spaces and X →→ E. Then the following imbedding are compact:

∂φ ∈ L1 (0, T ; Y ) →→ Lq (0, T ; E), if 1 ≤ q ≤ ∞; ∂t

∂φ ∞ ∈ Lr (0, T ; Y ) →→ C([0, T ]; E), if 1 < r ≤ ∞. φ : φ ∈ L (0, T ; X), ∂t

φ : φ ∈ Lq (0, T ; X),

3. Main results In this section, we state the main results on the hydrodynamic limit of the system (1.5). Before that, we need to give the formal derivation of the asymptotic model, obtain the basic energy estimate and introduce the relative entropy. 3.1. Formal derivation of the asymptotic model Denote  n (x, t) = 

 J (x, t) =





f (x, v, t) dv,

Rd

vf  (x, v, t) dv,

Rd

then  J  − n u =

(v − u )f  dv.

Rd

Firstly, integrating (1.5)1 with respect to v, yields ∂t n + divx J  = 0. Next, multiplying (1.5)1 by v and integrating with respect to v, yields  ∂t J  + divx

v ⊗ vf  dv = Rd

1 

 (u − v)f  dv, Rd

then simplifying the above formula as 1 ∂t J  + divx Q = (n u − J  ),  where  Q = 

Rd

Adding (3.1) and (1.5)3 , we obtain

v ⊗ vf  dv.

(3.1)

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∂t (ρ  u + J  ) + divx (ρ  u ⊗ u + Q ) + ∇x p  − x u = 0,

9

(3.2)

we decompose Q as follows:  Q = v ⊗ vf  dv Rd



=

        d ⊗ v f dv + u ⊗ vf dv − 2 ∇v f  ⊗ v f  dv, 

Rd

Rd

(3.3)

Rd

here   d  = (v − u ) f  + 2∇v f  . Then integrating the first term on right-hand side of (3.3) with respect to x and t, yields T  

 d  ⊗ v f  dvdxds

0  Rd

⎛ ⎞1 ⎛ ⎞1 2 2 T   T   ⎜ ⎟ ⎜ ⎟  2 2  ≤⎝ |d | dvdxds ⎠ ⎝ |v| f dvdxds ⎠ . 0  Rd

(3.4)

0  Rd

  Assuming that  Rd |v|√2 f  dvdx is bounded uniformly with respect to , then the above result can be controlled by O( ). In this process, we have used the fact that T   |d  |2 dvdxds ≤ C, 0  Rd

which can be obtained by the energy inequality. For the second term on right-hand side of (3.3), we have  u ⊗ vf  dv = u ⊗ J  .

(3.5)

Rd

Finally, for the third term on right-hand side of (3.3), one gets       2 ∇v f ⊗ v f dv = v ⊗ ∇v f  dv = −n I d. Rd

(3.6)

Rd

In short, (3.2) can be written as √ ∂t (ρ  u + J  ) + divx (ρ  u ⊗ u ) + divx (u ⊗ J  ) + ∇x (p  + n ) − x u = O( ). (3.7)

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Besides, T  J  − n u dxdt 0 

=

T   

f  d  dvdxdt

0  Rd

⎛ ⎜ ≤⎝

T  

⎞1 ⎛ 2

⎟ ⎜ f dvdxdt ⎠ ⎝

T  

2

⎟ |d | dvdxdt ⎠  2



0  Rd

⎞1

0  Rd

√

≤ C .

(3.8)

As a result, as  → 0, we have J  − n u → 0. We assume that ρ  , u converges to ρ, u respectively. Then we can pass to the limits of the nonlinear terms ρ  u and ρ  u ⊗ u , and obtain the corresponding limits ρu and ρu ⊗ u. Furthermore, J  → nu. So, formally, we have the following system of equations ⎧ ∂ n + divx (nu) = 0, ⎪ ⎪ t ⎪ ⎨ ∂ ρ + div (ρu) = 0, t x ⎪ ∂t ((ρ + n)u) + divx ((ρ + n)u ⊗ u) + ∇x P − x u = 0, ⎪ ⎪ ⎩ divx u = 0, here we denote P as n + p. 3.2. Basic energy estimate In this subsection, we consider a smooth solution (f  , ρ  , u ) of the problem (1.5)-(1.8). At first, integrating the kinetic equation (1.5)1 with respect to x and v over  × Rd , leads to the following equality: d dt

 

 f (x, v, t) dvdx +

(v · n (x))f  dSdv = 0,



 Rd

∂×Rd

and here the boundary term can be treated as follows:  (v · n (x))f  (x, v, t) dSdv ∂×Rd

 = v·

n(x)>0

 (v · n (x))f (x, v, t) dSdv + 

v·

n(x)<0

(v · n (x))f  (x, v, t) dSdv

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=

11

(v ∗ · n (x))f  (x, v ∗ , t) dSdv ∗

(v · n (x))f (x, v, t) dSdv − 

v ∗ ·

n(x)>0

v·

n(x)>0

= 0. Next, multiplying (1.5)1 by d dt

  

|v|2 2

+ ln(f  ) + 1 and integrating with respect to x and v, yields

|v|2  1 ( f + f  ln(f  )) dvdx − 2 

Rd

 

(u − v)f − ∇v f





+

(v · n (x))(

|v|2 2











  ∇v f  · v+ dvdx f

Rd

+ ln(f  ) + 1)f  dSdv = 0,

∂×Rd

where the boundary term is 0; we take the first boundary term as example 1 2

 (v · n (x))f  |v|2 dSdv ∂×Rd



1 = 2

1 (v · n (x))f |v| dSdv + 2 

v·

n(x)>0

(v · n (x))f  |v|2 dSdv v·

n(x)<0



1 = 2



2

(v · n (x))f  (x, v, t)|v|2 dSdv v·

n(x)>0

1 − 2



(v ∗ · n (x))f  (x, v ∗ , t)|v ∗ |2 dSdv ∗

v ∗ ·

n(x)>0

= 0. Furthermore, multiplying (1.5)3 by u and integrating with respect to x, yields d dt

 ρ

|u |2 dx + 2





  ∇x u 2 dx − 1 



  u · (v − u )f  dvdx = 0.  Rd

Finally, adding the above relations, we get ⎛

 

d ⎜ ⎝ dt

|v|2  ( f + f  ln(f  )) dvdx + 2

 Rd



+ 

where

  ∇x u 2 dx + 1 

 



⎞ |u |2 ⎟ ρ dx ⎠ 2



|d  |2 dvdx = 0, 

Rd

(3.9)

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1 

  |d  |2 dvdx  Rd

1 =− 

 



(u − v)f − ∇v f 







    ∇v f  1 u · (v − u )f  dvdx, · v+ dvdx − f 

 Rd

 Rd

√  f    √ √ −|v−u |2 −d and d  = (v − u ) f  + 2∇v f  = 2 M  ∇v M ). Setting  , M = (2π) 2 exp( 2   E(f , ρ , u ) = 





|v|2  ( f + f  ln(f  )) dvdx + 2

 Rd

 ρ

|u |2 dx. 2



Then integrating (3.9) with respect to t, leads to the following energy estimate: t  E(f , ρ , u )(t) + 





  ∇x u 2 dxds + 1 

0 

t   |d  |2 dvdxds = E(f  , ρ  , u )(0). (3.10) 0  Rd

3.3. Introduce the relative entropy Motivated by [15], we introduce the relative entropy   H (f , ρ , u ) = 





1 H (f |Mn,u ) dvdx + 2

 ρ  |u − u|2 dx.



 Rd



Next, we need to precise the expression of the relative entropy. Setting h(s) = s ln(s), we obtain H (f  |Mn,u ) = h(f  ) − h(Mn,u ) − h (Mn,u )(f  − Mn,u ) = f  ln(f  ) − Mn,u ln(Mn,u ) − (ln(Mn,u ) + 1)(f  − Mn,u )    f  = f ln + Mn,u − f  , Mn,u where Mn,u stands for the Maxwellian with density n and velocity u:   n(x, t) |v − u(x, t)|2 Mn,u = exp − . 2 (2π)d/2 Then integrating (3.11) with respect to x and v, yields   H (f  |Mn,u ) dvdx  Rd

(3.11)

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  =

 

 

f ln(f ) dvdx − 

f ln(Mn,u ) dvdx +



(Mn,u − f  ) dvdx



 Rd

13

 Rd

 Rd

= I1 + I 2 + I3 ,

(3.12)

where   I2 =

f  ln(Mn,u ) dvdx  Rd

 

=

  f ln 

  |v − u|2 n exp − 2 (2π)d/2

 Rd

 

=

d f ln(n) dvdx − 2

 

  f ln(2π) dvdx −



 Rd

dvdx



 Rd

f

|v − u|2 dvdx. 2

 Rd

For the term I3 , we have  

  Mn,u dvdx =



  n(x, t) |v − u(x, t)|2 exp − dvdx (2π)d/2 2

 Rd

Rd



=

n(x, t) dx. 

In view of the mass conservation, one follows  

  f dvdx = 

 Rd

 Rd

 

 n0 dx =

= 

here we used initial assumption ity. Consequently, we have



 n0

 

Mn,u dvdx,  Rd

dx =  

H (f |Mn,u ) dvdx = 

 Rd

f0 dvdx

   Rd

f0 dvdx in Theorem 3.1 in the second equal-

  |v − u|2 d  f ln(f ) + + ln(2π) dvdx 2 2 

 Rd



−

n ln(n) dx. 

In short, the relative entropy can be written as

(3.13)

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14

  H (f  , ρ  , u ) =

  |v − u|2 d f  ln(f  ) + + ln(2π) dvdx 2 2

 Rd



−



1 n ln(n) dx + 2

ρ  |u − u|2 dx.





(3.14)



3.4. Main theorems In this subsection, we state the main results about the hydrodynamic limit of the problem (1.5)-(1.8). Before that, we need to give the definition of weak solutions to this problem. Definition 3.1. We say (f  (x, v, t), ρ  (x, t), u (x, t)) is a global weak solution of the problem (1.5)-(1.8), if for any T > 0, the following properties hold: f  (x, v, t) ≥ 0, for any (x, v, t) ∈ ( × Rd × (0, T )); f  ∈ L∞ (0, T ; L1 ∩ L∞ ( × Rd )); |v|2 f  ∈ L∞ (0, T ; L1 ( × Rd )); ρ  ∈ L∞ (0, T ; L∞ ()); ρ  ∈ C([0, T ]; Lp ), 1 ≤ p < ∞; ρ  ≥ 0, for any (x, t) ∈  × (0, √ T ); • ρ  u ∈ L∞ (0, T ; L2 ()); u ∈ L2 (0, T ; H01 ()); • (1.5)2 and (1.5)3 hold in the sense of distribution, and (1.5)1 holds in the following sense:

• • • •



T   −

f 0 



 1  ∂t ϕ + v · ∇x ϕ + ((u − v) · ∇v ϕ + v ϕ) dvdxds 

Rd

 

=

f0 ϕ(x, v, 0) dvdx,  Rd

for any test function ϕ ∈ C ∞ ( × Rd × [0, T ]) such that ϕ(·, T ) = 0 and ϕ(x, v, t) = ϕ(x, v ∗ , t) for x ∈ ∂, v · n (x) > 0, (which holds in particular for test function ϕ independent of v). • the energy inequality  

|v|2  ( f + f  ln(f  )) dvdx + 2

 Rd

+

ρ

 |u

 |2

2

1 

≤

t  dx +

  ∇x u 2 dxds

0 



  



t   |d  |2 dvdxds 0  Rd

|v|2  ( f + f0 ln(f0 )) dvdx + 2 0

Rd

for any t ∈ [0, T ].

 ρ0 

|u0 |2 dx, 2

(3.15)

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15

For the main theorem, we need to give the following assumptions: (A1) ρ0 ≥ c(c > 0), ρ0 L∞ √() ≤ C, (A2) ρ0 − ρ0 L1 () = O( ),    2 |u |2 (A3)  Rd ( |v|2 f0 + f0 ln(f0 )) dvdx +  ρ0 20 dx < ∞, (A4) H (f0 , ρ0 , u0 ) → 0( → 0), (A5) ρ0 ≥ 0, ρ0 L∞ () ≤ C. At first, we state the hydrodynamic limit result of (1.5)-(1.8) when the initial density is bounded away from zero (ρ0 ≥ c). Theorem 3.1. Suppose that the initial data (f0 , ρ0 , u0 ) satisfy (1.6) and assumptions (A1)-(A4). Let (ρ0 , u0 , n0 ) be a smooth initial data for the limit problem (1.9) which satisfies 

  n0 dx =



f0 dvdx = C0 .  Rd

Let (ρ, u, n) be a smooth solution of the limit problem (1.9) on [0, T ]. Then there exists a positive constant C such that √ sup H (f  , ρ  , u )(t) ≤ CH (f  , ρ  , u )(0) + C .

(3.16)

0≤t≤T

Finally, we have sup H (f  , ρ  , u )(t) → 0, ( → 0). 0≤t≤T

Moreover, up to a subsequence, the weak solutions (f  , ρ  , u ) of the problem (1.5)-(1.8) satisfying inequality (3.16) satisfy: in L∞ (0, T ; L1 ( × Rd ),

f  → Mn,u u → u

in L2 (0, T ; Lq ()), 1 ≤ q ≤ 6,

ρ → ρ

in L∞ (0, T ; Lp ()), 1 ≤ p < ∞,

ρ  u → ρu n → n

in L2 (0, T ; L2 ()),

in L∞ (0, T ; L1 ()),

n u → nu

in L1 ( × (0, T )),

n u ⊗ u → nu ⊗ u

in L1 ( × (0, T )).

Remark 3.1. The global existence of weak solution to the initial-boundary value problem (1.5)-(1.8) can be done by using the similar arguments as in [25] and [30].

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16

Remark 3.2. The constant C depends on ∇x uL∞ , |G|2 L∞ , ∇x ρL2 () , F L∞ , ρ0 L∞ , xu and c is a positive lower bound of ρL∞ , T and c, here G = F − ∇x ln(n), F = ∇x (n+p)− ρ+n  ρ . The following result is about hydrodynamic limit of 2D inhomogeneous incompressible Navier-Stokes/Vlasov-Fokker-Planck equations when the initial density may vanish (ρ0 ≥ 0). Theorem 3.2. Suppose that the initial data (f0 , ρ0 , u0 ) satisfy (1.6) and assumptions (A2)-(A5). Let (ρ0 , u0 , n0 ) be a smooth initial data for the limit problem (1.9) which satisfies    n0 dx = f0 dvdx = C0 . 

 R2

Let (ρ, u, n) be a smooth solution of the limit problem (1.9) on [0, T ]. Then there exists a positive constant C and a time T1 such that √   sup H (f  , ρ  , u )(t) ≤ CH (f  , ρ  , u )(0) + CK41 + CK42 + C ,

(3.17)

0≤t≤T1

 + K  → 0( → 0), for all t ≤ T . Finally, we have with K41 1 42

sup H (f  , ρ  , u )(t) → 0, ( → 0). 0≤t≤T1

Moreover, up to a subsequence, the weak solutions (f  , ρ  , u ) of the problem (1.5)-(1.8) satisfying inequality (3.16) satisfy: in L∞ (0, T1 ; L1 ( × R2 ),

f  → Mn,u u → u

in L2 (0, T1 ; Lq ()), 1 ≤ q < ∞,

ρ → ρ

in L∞ (0, T1 ; Lp ()), 1 ≤ p < ∞,

ρ  u → ρu n → n

in L2 (0, T1 ; L2 ()),

in L∞ (0, T1 ; L1 ()),

n u → nu

in L1 ( × (0, T1 )),

n u ⊗ u → nu ⊗ u

in L1 ( × (0, T1 )).

 and K  are defined in (7.11); The constant C depends on ∇ u ∞ , Remark 3.3. K41 x L 42 2 |G| L∞ , ∇x ρL2 () , F L∞ , ρ0 L∞ , ρL∞ and T1 ; T1 ≤ 4C(F  ∞ ,ρ   ∞1 ,ρ ∞ ,∇x ρ 2 ) , L L 0 L L see (7.12).

Remark 3.4. Theorem 3.2 doesn’t hold for 3D case. In the proof of Theorem 3.2, we use a compactness results LM →→ H −1 of Orlicz space in 2D to obtain the convergence of n → n in L∞ (0, T1 ; H −1 ()), which is used to estimate K4 . The compactness result is LM →→ 3 H − 2 in 3D, and we can not obtain the convergence of n , thus we have no method to deal with K4 in 3D when ρ0 ≥ 0.

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17

Remark 3.5. The method in this paper can be used to prove the hydrodynamic limit of weak solutions to the compressible Navier-Stokes-Vlasov equations without diffusive term v f  in one dimension (see [9]). 4. A priori estimates Proposition 4.1. Let (f  , ρ  , u ) be a solution of the problem (1.5) − (1.8), which satisfies the assumption (A3), then the following results holds: • u is bounded in L2 (0, T ; H01 ()); • f  (1 + |v|2 + | ln(f  )|) is bounded in L∞ (0, T ; L1 ( × Rd )); • √1 d  is bounded in L2 ( × Rd × (0, T )). Proof. Firstly, by the total mass is conserved  

  f  (x, v, t) dvdx =

 Rd

f0 (x, v) dvdx,  Rd

one gets f  is bounded in L∞ (0, T ; L1 ( × Rd )). Next, we deduce the estimate of f  | ln(f  )|. Introducing s| ln(s)| = s ln(s) − 2s ln(s)χ0≤s≤1 .

(4.1)

For ω ≥ 0, we have −s ln(s)χ0≤s≤1 = −s ln(s)χe−ω ≤s≤1 − s ln(s)χe−ω ≥s √ ≤ sω + C sχe−ω ≥s ≤ sω + Ce−ω/2 , setting s = f  , ω =

|v|2 8 ,

one gets

 

  f | ln(f )| dvdx ≤ 



 Rd

f  ln(f  ) dvdx  Rd

+

1 4

 

  |v|2 f  dvdx + 2C

 Rd

e−

|v|2 16

dvdx,

 Rd

combining this formula with the energy estimate (3.10), we obtain  

1 f (1 + | ln(f )|) dvdx + 4 

 

 Rd

1 |v| f dvdx + 2

 Rd

T  + 0 

  ∇x u 2 dxds + 1 

 ρ  |u |2 dx

2 





T   |d  |2 dvdxds 0  Rd

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18

T 

  ∇x u 2 dxds + 1 

≤E(f , ρ , u )(t) + 





0 

T   |d  |2 dvdxds + C 0  Rd

⎛

⎜ ≤E(f  , ρ  , u )(0) + C ⎝1 +

T  

⎞

⎟ |v|2 f  dvdxds ⎠ ,

0  Rd

by Gronwall’s inequality, we have   |v|2 f  dvdx ≤ C.  Rd

Furthermore, we obtain u is bounded in L2 (0, T ; H01 ()), (0, T )). 2

√1 d  

is bounded in L2 ( × Rd ×

Proposition 4.2. Under the hypotheses of Proposition 4.1, the following assertion holds |v − u |2 f ε is bounded in L1 ( × Rd × (0, T )). Proof. We note that  |v − u |2 f  dv Rd

 =

   |d  |2 − 4|∇v f  |2 − 4∇v f  · (v − u ) f  dv

Rd





|d | dv + 2

≤

(v − u ) · ∇v f  dv

 2

Rd





|d  |2 + 2

= Rd

Rd

v · ∇v f  dv

Rd





|d  |2 dv + 4

≤ Rd

f  dv,

Rd

integrating the above inequality with respect to x and t , yields T   |v − u |2 f  dvdxds 0  Rd

T  

T   |d | dvdxds + 4

≤

 2

0  Rd

f  dvdxds 0  Rd

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19

≤ C() + C(T ), where we have used Proposition 4.1. Then, we obtain |v − u |2 f ε is bounded in L1 ( × Rd × (0, T )). 2 Proposition 4.3. Under the hypotheses of Proposition 4.1, the following assertions holds • • • •

nε is bounded in L∞ (0, T ; L1 ()); nε |u |2 and nε u is bounded in L1 ( × (0, T )); √ J  − n u , Q − n (I + u ⊗ u ), and (J  − n u ) ⊗ u are O( ) in L1 ( × (0, T )); nε | ln(nε )| is bounded in L∞ (0, T ; L1 ()).

Proof. From the total mass conserved, we obtain the first result. Next, by the Cauchy inequality, we get T  nε |u |2 dxds 0 

T   f  |u |2 dvdxds

= 0  Rd

T   =

f  |u − v + v|2 dvdxds 0  Rd

T   ≤2

T   |u − v| f dvdxds + 2 

|v|2 f  dvdxds

2 

0  Rd

0  Rd

≤ C, where we have used Proposition 4.1 and Proposition 4.2. Similarly, we have T 

⎛ T ⎞ 12 ⎛ T ⎞ 12     nε u dxds ≤ ⎝ n dxds ⎠ ⎝ nε |u |2 dxds ⎠

0 

≤

0 

1 2

T

 n dxds +

0 

≤ C. Furthermore,

1 2

0 

T

 n |u |2 dxds

0 

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20

T  Q − n (I + u ⊗ u ) dxds 0 

T   =

(v ⊗ v − I − u ⊗ u )f  dvdxds 0  Rd

T   =

    (d  ⊗ v f  + u f  ⊗ d  − If  − 2∇v f  ⊗ v f 

0  Rd

  − 2u f  ⊗ ∇v f  ) dvdxds T   =

  d  ⊗ v f  + u f  ⊗ d  dvdxds

0  Rd

⎛ ⎜ ≤⎝

⎞1 ⎛

T  

2

⎟ ⎜ |d | dvdxds ⎠ ⎝

T  

 2

0  Rd

⎞1

2

⎟ (|v| + |u | ))f dvdxds ⎠ 2

 2



0  Rd

√

≤C . Finally, because of  (J − n u ) ⊗ u = 

 

(v − u ) ⊗ u f  dv



Rd



=

  (v − u ) f  ⊗ u f  dv

Rd



=

     ((v − u ) f  + 2∇v f  ) ⊗ u f  − 2∇v f  ⊗ u f  dv

Rd



=

 d  ⊗ u f  dv,

Rd

then integrating this inequality with respect to x and t , yields T  (J  − n u ) ⊗ u dxds 0 

⎛ ⎞1 ⎛ ⎞1 2 2 T   T   ⎜ ⎟ ⎟ ⎜ ≤⎝ |d  |2 dvdxds ⎠ ⎝ |u |2 f  dvdxds ⎠ 0  Rd

0  Rd

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21

√ ≤ C . Finally, because h(s) = s ln(s) is convex, the Jensen inequality holds: ⎛ ⎜ h(n ) = h ⎝

⎞



f



M1

⎟ M1 dv ⎠ ≤

Rd

where M1 = |v|2 2 )

dv +

d 2

1 (2π)d/2





 f h M1 dv, M1

Rd

exp{− |v|2 }. The right hand-side of the above inequality is 2

 Rd

f  (ln(f  ) +

ln(2π)n , which is bounded in L∞ (0, T ; L1 ()). By using (4.1), we have  nε | ln(nε )| dx ≤ C. 

We complete the proof of Proposition 4.3.

2

5. Relative entropy estimate In this section, we give the relative entropy estimate. At first, we have the following lemma: Lemma 5.1. The relative entropy defined by (3.14) satisfies 1 H (f , ρ , u )(t) +  



t  

t  |d | dvdxds +  2



0  Rd

  ∇x (u − u)2 dxds

0 

t   ≤H (f , ρ , u )(0) − 



f  (v − u) ⊗ (v − u) : ∇x u dvdxds



0  Rd

t   +

(F − ∇x ln(n)) · (v − u )f  dvdxds 0  Rd

t  +

(n − n)(u − u) · (F − ∇x ln(n)) dxds 0 

t  −

t  ρ (u − u) ⊗ (u − u) : ∇x u dxds + 

0 



(ρ  − ρ)(u − u) · F dxds,



0 

(5.1) where F =

∇x (n+p)−x u . ρ+n

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22

Proof. First, since (1.5)2 holds in the sense of distribution, the integral identity 

t 

 ρ ϕ(·, t) dx = 



ρ0 ϕ(·, 0)

dx +

ρ  ∂t ϕ + ρ  u · ∇x ϕ dxds

(5.2)

0 



holds for any test function ϕ ∈ Cc∞ ( × [0, T ]). Taking ϕ = 12 |u|2 , we have 

1  2 ρ |u| (·, t) dx = 2





1  ρ |u0 |2 dx + 2 0

t  ρ  u · ∂t u + ρ  u · ∇x u · u dxds.

(5.3)

0 



Next, (1.5)3 holds in the sense of distribution, the integral identity 

t 

 (ρ u ) · φ(·, t) dx =  



ρ0 u0

· φ(·, 0) dx +

ρ  u · ∂t φ + ρ  u ⊗ u : ∇x φ 0 



⎛

⎜1 + p  divx φ − ∇x u : ∇x φ + ⎝ 



⎞ ⎟ (v − u )f  · φ dv ⎠ dxds (5.4)

Rd

holds for any test function φ ∈ Cc∞ ( × [0, T ]). Taking φ = u, we obtain 

t 

 ρ u · u(·, t) dx =  



ρ0 u0

· u0 dx +

ρ  u · ∂t u + ρ  u ⊗ u : ∇x u 0 



⎛



⎜1 − ∇x u : ∇x u + ⎝ 

⎞ ⎟ (v − u ) · uf  dv ⎠ dxds.

(5.5)

Rd

Finally, (1.5)1 holds in the sense of distribution, then  

  f ψ(·, ·, t) dvdx −

f0 ψ(·, ·, 0) dvdx



 Rd

 Rd



t   =

f



1 ∂t ψ + v · ∇x ψ + ((u − v) · ∇v ψ + v ψ 

 dvdxds,

(5.6)

0  Rd

for any test function ψ ∈ C ∞ ( × Rd × [0, T ]). Taking ψ = yields

|u|2 2

−v·u+

d 2

ln(2π) − ln(n),

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  f  Rd

23

 d |u|2 − v · u + ln(2π) − ln(n) dvdx 2 2  d |u0 |2 − v · u0 + ln(2π) − ln(n0 ) dvdx 2 2



 

−

f0  Rd



t   =



f 0 

∂t n u · ∂t u − v · ∂t u − n

 dvdxds

Rd

  |u|2 f  v · ∇x − v · ∇x u − v · ∇x ln(n) dvdxds 2

t   + 0  Rd

t  

1   f (u − v) · (−u) dvdxds. 

+

(5.7)

0  Rd

Combine with (5.3), (5.5), (5.7) and energy inequality (3.15), using the condition divu = 0 and the limit equation, we have 1 H (f , ρ , u )(t) +  



t  

t  |d | dvdxds +  2



0  Rd

  ∇x (u − u)2 dxds

0 

t

 

≤H (f  , ρ  , u )(0) −

f  (v − u) ⊗ (v − u) : ∇x u dvdxds 0  Rd

t   (F − ∇x ln(n)) · (v − u )f  dvdxds

+ 0  Rd

t  +

(n − n)(u − u) · (F − ∇x ln(n)) dxds 0 

t  ρ  (u − u) ⊗ (u − u) : ∇x u dxds

− 0 

t  (ρ  − ρ)(u − u) · F dxds

+ 0 

=

6 ! i=1

Ki .

(5.8)

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24

Hence, we complete the proof of Lemma 5.1.

2

To complete the proof of Theorem 3.1, we need to derive the estimates of Ki (i = 1, · · · , 6). Firstly, for K5 , we obtain the following inequality:  t           |K5 | =  ρ (u − u) ⊗ (u − u) : ∇x u dxds    0 

t

 ∇x uL∞

≤ 0

ρ  |u − u|2 dxds 

t ≤C

H (f  , ρ  , u ) ds, 0

where C depends on ∇x uL∞ . Next, for K3 , we have the following estimate: t   (F − ∇x ln n) · (v − u )f  dvdxds 0  Rd

t   =

t           [(v − u ) f + 2∇v f ] · G f dvdxds − 2∇v f  · G f  dvdxds 

0  Rd

t

 

=

0  Rd

 d  · G f  dvdxds

0  Rd

⎛ ⎜ ≤⎝

t  

⎞1 ⎛ 2

⎟ ⎜ |d | dvdxds ⎠ ⎝

t  

 2

0  Rd

√

⎞1

2

⎟ |G| f dvdxds ⎠ 2 

0  Rd

≤ C , where G = F − ∇x ln(n), and C depends on |G|2 L∞ . Furthermore, we estimate K2 . Owing to (v − u) ⊗ (v − u) = (v − u + u − u) ⊗ (v − u + u − u) = (v − u ) ⊗ (v − u ) + (v − u ) ⊗ (u − u) + (u − u) ⊗ (u − u) + (u − u) ⊗ (v − u ). Then K2 can be split into four parts denoted by K2i (i = 1, 2, 3, 4), we have

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25

t   K21 =

f  (v − u ) ⊗ (v − u ) : ∇x u dvdxds 0  Rd

=

t   

 f  (v − u ) ⊗ (v − u ) f  : ∇x u dvdxds

0  Rd

t   =

  (d  − 2∇v f  ) ⊗ (v − u ) f  : ∇x u dvdxds

0  Rd

t   =

 d  ⊗ (v − u ) f  : ∇x u dvdxds

0  Rd

⎛ ⎞1 ⎛ ⎞1 2 2 t   t   ⎜ ⎟ ⎜ ⎟ ≤⎝ |d  |2 dvdxds ⎠ ⎝ (|v|2 + |u |2 )f  dvdxds ⎠ 0  Rd

0  Rd

√

≤ C , and t   K22 =

f  (v − u ) ⊗ (u − u) : ∇x u dvdxds 0  Rd

t   =

  (d  − 2∇v f  ) ⊗ (u − u) f  : ∇x u dvdxds

0  Rd

t   =

 d  ⊗ (u − u) f  : ∇x u dvdxds

0  Rd

⎛ ⎜ ≤⎝

t  

⎞1 ⎛ 2

⎟ ⎜ |d | dvdxds ⎠ ⎝

t  

 2

0  Rd

√

⎞1

2

⎟ (|u| + |u | )f dvdxds ⎠ 2

 2



0  Rd

≤ C , where t   0  Rd

Similarly,

  2∇v f  ⊗ (u − u) f  : ∇x u dvdxds = 0.

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26

√ K24 ≤ C . Furthermore t   K23 =

f  (u − u) ⊗ (u − u) : ∇x u dvdxds 0  Rd

t   f  |u − u|2 dvdxds,

≤ ∇x uL∞ 0  Rd

and t   f  |u − u|2 dvdxds 0  Rd

t  =

n |u − u|2 dxds 0 

t   =

(|u − u|2 − |v − u|2 + |v − u |2 )f  dvdxds 0  Rd

t   +

(2 ln(f  ) + |v − u|2 )f  dvdxds 0  Rd

t   (2 ln(f  ) + |v − u |2 )f  dvdxds

− 0  Rd

t   (u − u) · (v − u )f  dvdxds

=−2 0  Rd

t   +2

(H (f  |Mn,u ) − H (f  |Mn,u )) dvdxds 0  Rd

t    "   # f  (u − u) · (v − u ) f  + 2∇v f  dvdxds ≤−2 0  Rd

t    t       f (u − u) · 2∇v f dvdxds + 2 H (f  |Mn,u ) dvdxds +2 0  Rd

0  Rd

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⎛ ⎜ ≤C ⎝

t  

⎞1 ⎛ 2

⎟ ⎜ |d | dvdxds ⎠ ⎝

t  

 2

0  Rd

27

⎞1

2

⎟ (|u| + |u | )f dvdxds ⎠ 2

 2



0  Rd

t   +2

H (f  |Mn,u ) dvdxds 0  Rd

t  

√

≤C  + 2

H (f  |Mn,u ) dvdxds,

(5.9)

0  Rd

where we have used the fact H (f  |Mn,u ) ≥ 0. On the other hand, since for any 0 ≤ α ≤ β, we have measure {x ∈ |α ≤ ρ  (x, t) ≤ β} does not change in term of time t . See Lions’book [23] (Page 23, Theorem 2.1). Then when the initial data ρ0 (x) ≥ c, one follows ρ  (x, t) ≥ c. Furthermore, for K4 , we get the following result: t  K4 =

(n − n)(u − u) · (F − ∇x ln(n)) dxds 0 

t  ≤ G

|n − n||u − u| dxds

L∞ 0 

⎛ t ⎜ ≤ GL∞ ⎝



t

⎞



⎟ |n − n||u − u| dxds ⎠ ,

|n − n||u − u| dxds +

0 |n−n |≤n

0 |n−n |≥n

where G = F − ∇x ln(n). By using Lemma 2.2 and the fact that ρ  ≥ c, we obtain t

 |n − n||u − u| dxds

0 |n−n |≤n

1 ≤ 2c C ≤ 2c

t



c |n − n| dxds + 2

0 |n−n |≤n

t



t

 |u − u|2 dxds

2



0 |n−n |≤n

1 nH (n |n) dxds + 2

t

 ρ  |u − u|2 dxds



0 |n−n |≤n

C  nL∞ ≤ 2c

t 

1 H (n |n) dxds + 2

0 |n−n |≤n

t

 ρ  |u − u|2 dxds



0 

0 |n−n |≤n

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28

t   ≤K

1 H (f |Mn,u ) dvdxds + 2

t  ρ  |u − u|2 dxds



0  Rd

 ≤K

0 

t H (f  , ρ  , u ) ds. 0

Besides, we have t

 |n − n||u − u| dxds

0 |n−n |≥n

t



t

≤



|n − n||u − u| dxds + 

0 |n−n |≥n,|u −u|≤1

t

0 |n−n |≥n,|u −u|≥1



t 

≤

|n − n| dxds +

|n − n||u − u|2 dxds



0 |n−n |≥n

 ≤C

t

0 



t  H (n |n) dxds + nL∞

0 

 ≤C

0 

 

0  Rd

nL∞ c

n |u − u|2 dxds

2



H (f  |Mn,u ) dvdxds + t

t  |u − u| dxds +



t

|n − n||u − u| dxds



0 

t

 ρ  |u − u|2 dxds

0 



+

n |u − u|2 dxds. 0 

The last integral in the right hand of the above inequality is the same as (5.9), and one obtains  K4 ≤ K

t

√ H (f  , ρ  , u ) ds + C .

(5.10)

0

Since (1.5)2 holds in the sense of distribution, combining with the limit equation ∂t ρ + divx (ρu) = 0, we have ∂t (ρ  − ρ) + (u − u) · ∇x ρ + div(u (ρ  − ρ)) = 0, in D ( × [0, T ]). Fix small constants δ1 > 0, δ2 > 0, for any t ≤ T , choose

(5.11)

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29

ψδ 1 ∈ D(0, t), ψδ 1 → 1 as δ1 → 0; ψδ 1 = 1 for τ ≥ δ1 or τ ≤ t − δ1 ; φδ 2 ∈ D(), φδ 2 → 1 as δ2 → 0; φδ 2 = 1 for x ∈ , dist(x, ∂) ≥ δ2 and |∇x φδ 2 | ≤

2 . δ2

Take ψδ 1 (t)φδ 2 (x) as test function of (5.11), one obtains 

t 

 (ρ  − ρ)ψδ 1 (t)φδ 2 (x) dx =



(ρ0 − ρ0 )ψδ 1 (0)φδ 2 (x) dx +

(ρ  − ρ)∂t ψδ 1 φδ 2 dxds 0 



t  −

(u − u) · ∇x ρ ψδ 1 φδ 2 dxds 0 

t  +

u (ρ  − ρ) · ∇x φδ 2 ψδ 1 dxds.

(5.12)

0 

Next, we estimate  t the  terms in (5.12) one by one, and send δ2 → 0, δ1 → 0. Here we mainly discuss the term 0  u (ρ  − ρ) · ∇x φδ 2 ψδ 1 dxds. For the other four terms, it is easy to deal with by using the properties of ψδ 1 and φδ 2 .   t          u (ρ − ρ) · ∇ φ ψ dxds x δ2 δ1     0 

t  ≤

(ρ  − ρ)|u |[dist(x, ∂)]−1 [dist(x, ∂)]|∇x φδ 2 |ψδ 1 dxds

0 

t



t

≤C



|ρ − ρ| dxds ≤ C 

|ρ  |2 + |ρ|2 dxds → 0, δ2 → 0,

2

0 dist(x,∂)≤δ2

0 dist(x,∂)≤δ2

where we used the Hardy-type inequality to obtain |u |[dist(x, ∂)]−1 ∈ L2 (0, T ; L2 ()), and ρ L∞ () ≤ C. At last, we have 

t 

 (ρ − ρ) dx = 



(ρ0 

− ρ0 ) dx −

(u − u) · ∇x ρ dxds,

(5.13)

0 

and 

t |ρ − ρ| dx 



≤ρ0

− ρ0 L1 () + C(∇x ρL2 )

u − uL2 () ds. 0

(5.14)

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30

Thus, there exists a positive constant C depending only on ∇x ρL2 , F L∞ , ρ0 L∞ , ρL∞ , T and c (c is a positive lower bound of ρ  ) such that t  K6 =

(ρ  − ρ)(u − u) · F dxds 0 

t ≤C(F L∞ )

ρ  − ρL2 () u − uL2 () ds 0

t ≤C(F L∞ , ρ0 L∞ , ρL∞ )

ρ  − ρL1 () u − uL2 () ds 0

t ≤C(F L∞ , ρ0 L∞ , ρL∞ )

ρ0 − ρ0 L1 () u − uL2 () ds 0

⎞ ⎛ s  ⎝ u − uL2 () dr ⎠ u − uL2 () ds

t + C(F L∞ , ρ0 L∞ , ρL∞ , ∇x ρL2 ) ⎛ 1 ≤C(F L∞ , ρ0 L∞ , ρL∞ ) ⎝ 2c

0

t 0

0

c ρ0 − ρ0 2L1 () + 2

t 

⎞ |u − u|2 dxds ⎠

0 

⎛

c + C(F L∞ , ρ0 L∞ , ρL∞ , ∇x ρL2 ) ⎝ 2

t



⎞

|u − u|2 dx ds ⎠

0 

1 + C(F L∞ , ρ0 L∞ , ρL∞ , ∇x ρL2 ) 2c

t

⎛ s  ⎝ u − u2 2

L ()

0

t

⎞2 dr ⎠ ds

0



≤C(F L∞ , ∇x ρL2 , ρ0 L∞ , ρL∞ , T , c)

ρ  |u − u|2 dxds + C 0 

t ≤C(F 

L∞

, ∇x ρL2 , ρ0 L∞ , ρL∞ , T , c)

H (f  , ρ  , u ) ds + C 0

t ≤C

H (f  , ρ  , u ) ds + C, 0

where we have used assumption (A2). Then combine the above results, we have

(5.15)

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1 H (f , ρ , u )(t) +  



t  

t  |d | dvdxds +  2



0  R2

31

  ∇x (u − u)2 dxds

0 

t ≤H (f  , ρ  , u )(0) + C

√ H (f  , ρ  , u )ds + C .

(5.16)

0

In the end, by Gronwall’s inequality, it holds that √ sup H (f  , ρ  , u )(t) ≤ CH (f  , ρ  , u )(0) + C .

(5.17)

0≤t≤T

Thus, we complete the proof of (3.16). 6. Passing to the limit In this section, we need to obtain the convergence of the macroscopic quantities and microscopic quantities to complete the proof of Theorem 3.1. At first, from (5.16), (5.17) and the Poincaré inequality, which gives u → u

in L2 (0, T ; Lq ()), 1 ≤ q ≤ 6.

(6.1)

By (5.14), (6.1) and assumption (A2), we get ρ → ρ

in L∞ (0, T ; Lp ()), 1 ≤ p < ∞.

(6.2)

Furthermore, one follows (p = q = 4) ρ  u → ρu in L2 (0, T ; L2 ()).

(6.3)

ρ  u ⊗ u → ρu ⊗ u in L1 ( × (0, T )).

(6.4)

Then we have

Indeed, T 

    ρ u ⊗ u − ρu ⊗ u dxdt

0 

T  =

    ρ (u − u) ⊗ (u − u) + ρ  u ⊗ u + ρ  u ⊗ u − ρ  u ⊗ u − ρu ⊗ u dxdt

0 

T  ρ  |u − u|2 + 2|u||ρ  u − ρu| + |ρ  − ρ||u|2 dxdt

≤ 0 

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32

T ≤

H (f  , ρ  , u )(t) dt + CuL2 (0,T ;L2 ()) ρ  u − ρuL2 (0,T ;L2 ()) 0

+ Cρ  − ρL2 (0,T ;Lp ()) |u|2 L∞ (0,T ;L∞ ()) → 0, ( → 0), where the limit solution u is smooth enough. For the convergence of microscopic quantities, by   H (f  |Mn,u ) dvdx → 0, as  → 0,  R2

and since f  − Mn,u 2L1 (×Rd ) is dominated by f  → Mn,u

   R2

H (f  |Mn,u ) dvdx, then we have

in L∞ (0, T ; L1 ( × Rd )).

(6.5)

Then Combining Lemma 2.2, Csiszár-Kullback-Pinsker inequality and the strong convergence of f  above, one has n → n strongly in L∞ (0, T ; L1 ()). Furthermore, we consider the convergence of the nonlinear terms n u and n u ⊗ u . At first, from (5.9) and (5.17), we obtain T  |n u − nu| dxds 0 

T  =

T  |n (u − u)| dxds + 

|(n − n)u| dxds



0 

0 

⎞ 12 ⎛ T ⎞ 12 ⎛ T     n dx ds ⎠ ⎝ n |u − u|2 dxds ⎠ ≤⎝ 0 

0 

+ Cn − nL∞ (0,T ;L1 ()) → 0, ( → 0), 

therefore, n u → nu in L1 (0, T ; L1 ()). Next, we have

(6.6)

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T  |n u ⊗ u − nu ⊗ u| dxds 0 

T  ≤

T  |n (u − u) ⊗ u | dxds + 



0 

T

|n u ⊗ (u − u)| dxds



0 

 |(n − n)u ⊗ u| dxds

+ 0 

=E1 + E2 + E3 . By Proposition 4.3, Hölder’s inequality and (5.9), one obtains ⎛ T ⎞ 12 ⎛ T ⎞ 12     E1 ≤ ⎝ n |u |2 dxds ⎠ ⎝ n |u − u|2 dxds ⎠ 0 

0 

⎛

T

√

≤C⎝ +

⎞ 12 H (f  , ρ  , u ) ds ⎠ → 0, ( → 0),

0

and ⎞ 12 ⎛ T ⎞ 12 ⎛ T     E2 ≤ uL∞ ⎝ n |u − u|2 dxds ⎠ ⎝ n dxds ⎠ 0 

⎛ ≤C⎝

T 

0 

⎞

1 2

n |u − u|2 dxds ⎠

0 

⎛

√ ≤C⎝ +

T

⎞ 12 H (f  , ρ  , u ) ds ⎠ → 0, ( → 0).

0

For E3 , we have T E3 ≤

n − nL1 () |u2 |L∞ () ds 0

≤ Cn − nL∞ (0,T ;L1 ()) → 0, ( → 0).

33

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Thus, n u ⊗ u → nu ⊗ u

in L1 (0, T ; L1 ()).

(6.7)

In the end, we complete the proof of Theorem 3.1. 7. Proof of Theorem 3.2 The aim of this section is to complete the proof of Theorem 3.2. In fact, ρ0 ≥ c in Theorem 3.1 is only used in the estimates of K4 and K6 . Thus, to prove Theorem 3.2, we only need to deal with estimate K4 and K6 when ρ0 ≥ 0. In order to estimate K4 , we firstly prove the convergence of n . Using Proposition 4.1, we have u is bounded in L2 (0, T ; H01 ()),

(7.1)

then there exists a subsequence (still denoted by u ) such that u  u weakly in L2 (0, T ; H01 ()).

(7.2)

Next, using Proposition 4.3, we obtain n is bounded in L∞ (0, T ; L1 ()).

(7.3)

Integrating the kinetic equation (1.5)1 with respect to v, we have ∂t n + divx J  = 0.

(7.4)

n u is bounded in L1 ( × (0, T )),

(7.5)

J  − n u is bounded in L1 ( × (0, T )).

(7.6)

J  is bounded in L1 ( × (0, T )).

(7.7)

∂t n is bounded in L1 (0, T ; W −1,1 ()).

(7.8)

From Proposition 4.3, we obtain

and

Then, one follows

Furthermore, by (7.4), we get

Besides, from Proposition 4.3, we have  nε | ln nε | dx ≤ C, 

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35

which implies  (1 + nε ) ln(1 + nε ) dx ≤ C. 

   This together with mass conservation of n (  n dx =  Rd f0 dvdx = C0 ) imply n is bounded in L∞ (0, T ; LM ()).

(7.9)

Thus, applying Lions-Aubin lemma (Lemma 2.3), we obtain n → n

in L∞ (0, T ; H −1 ()),

(7.10)

where we have used the compact embedding in 2D (Lemma 2.1) LM () →→ H −1 (). Finally, we get the following result: t  K4 =

(n − n)(u − u) · (F − ∇x ln(n)) dxds 0 

t 

t  (n − n)u · (F − ∇x ln(n)) dxds −

= 0   = K41

(n − n)u · (F − ∇x ln(n)) dxds 0 

 + K42

→ 0, as  → 0,

(7.11)

where we have used the fact that (n, ρ, (n + ρ)u) is a smooth solution of the asymptotic system (1.9), we treat u · (F − ∇x ln(n)) and u · (F − ∇x ln(n)) as the elements in the dual space of L∞ (0, T ; H −1 ()). Furthermore, K6 can be estimated as follows: t  K6 =

(ρ  − ρ)(u − u) · F dxds 0 

t ρ  − ρL2 () u − uL2 () ds

≤C(F L∞ ) 0

t ≤C(F L∞ , ρ0 L∞ , ρL∞ )

ρ  − ρL1 () u − uL2 () ds 0

t ≤C(F L∞ , ρ0 L∞ , ρL∞ )

ρ0 − ρ0 L1 () u − uL2 () ds 0

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36

t + C(F L∞ , ρ0 L∞ , ρL∞ , ∇x ρL2 ) ⎛ 1 ≤C(F L∞ , ρ0 L∞ , ρL∞ ) ⎝ 4σ

⎞ ⎛ s  ⎝ u − uL2 () dr ⎠ u − uL2 () ds

0

0

t

t  ρ0 − ρ0 2L1 () + σ

0

⎛

+ C(F L∞ , ρ0 L∞ , ρL∞ , ∇x ρL2 ) ⎝σ

⎞ |u − u|2 dxds ⎠

0 

t 

⎞

|u − u|2 dx ds ⎠

0 

1 + C(F L∞ , ρ0 L∞ , ρL∞ , ∇x ρL2 ) 4σ

t

⎛ s  ⎝ u − u2 2

L ()

0

1 ≤C(F L∞ , ρ0 L∞ , ρL∞ , ∇x ρL2 ) t 2 σ

t

⎞2 dr ⎠ ds

0

 |∇x u − ∇x u|2 dxds + C

0 

t  + C(F L∞ , ρ0 L∞ , ρL∞ , ∇x ρL2 )σ

|∇x (u − u)|2 dxds 0 

≤

1 2

t

 |∇x u − ∇x u|2 dxds + C,

(7.12)

0 

where we have used assumption (A2), the Poincaré inequality and we also take σ = 1 1 4C(F L∞ ,ρ0 L∞ ,ρL∞ ,∇x ρL2 ) and t ≤ T1 ≤ 4C(F L∞ ,ρ0 L∞ ,ρL∞ ,∇x ρL2 ) . Then, combining the estimates of K1 -K3 , K5 in Section 5 and K4 , K6 above, we have 1 H (f , ρ , u )(t) +  



t  

t  |d | dvdxds +  2



0  R2

  ∇x (u − u)2 dxds

0 

t ≤H (f , ρ , u 





 )(0) + K41

 + K42

+C

√ H (f  , ρ  , u )ds + C , for t ≤ T1 .

0

(7.13) By Gronwall’s inequality, it holds that √   sup H (f  , ρ  , u )(t) ≤ CH (f  , ρ  , u )(0) + CK41 + CK42 + C  → 0, as  → 0. (7.14)

0≤t≤T1

Similar to (6.5)-(6.6), (7.14) and Csiszár-Kullback-Pinsker inequality imply n → n strongly in L∞ (0, T1 ; L1 ()).

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