Global weak solutions to 3D compressible Navier–Stokes–Poisson equations with density-dependent viscosity

Accepted Manuscript Global weak solutions to 3D compressible Navier-Stokes-Poisson equations with density-dependent viscosity

Yulin Ye, Changsheng Dou

PII: DOI: Reference:

S0022-247X(17)30504-8 http://dx.doi.org/10.1016/j.jmaa.2017.05.044 YJMAA 21405

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

18 March 2017

Please cite this article in press as: Y. Ye, C. Dou, Global weak solutions to 3D compressible Navier-Stokes-Poisson equations with density-dependent viscosity, J. Math. Anal. Appl. (2017), http://dx.doi.org/10.1016/j.jmaa.2017.05.044

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Global weak solutions to 3D compressible Navier-Stokes-Poisson equations with density-dependent viscosity Yulin Ye∗

and Changsheng Dou† May 19, 2017

∗ School

† School

of Mathematics and Statistics, Henan University, Kaifeng 475004, P.R. China. Email: [email protected]

of Statistics, Capital University of Economics and Business, Beijing 100070, P.R. China. Email: [email protected]

Abstract: In this paper, we study the global existence of weak solutions to the compressible Navier-Stokes-Poisson (N-S-P) equations with density-dependent viscosity and non-monotone pressure in a three dimensional torus. Our approach is based on the FaedoGalerkin method and the compactness arguments. Motivated by Vasseur-Yu [30] and [31], we construct the approximate solutions and the key estimates ling in the elementary energy estimates, B-D entropy and Mellet-Vasseur type inequality for the weak solutions. Here, we need the conditions that the adiabatic constant γ satisfies 43 < γ < 3, for λ = −1 or 1 < γ < 3, for λ = 1, where λ is a sign constant of Poisson equation which determines the physical meaning of the N-S-P system. Keywords: compressible Navier-Stokes-Poisson equations; density-dependent viscosity; global weak solutions; vacuum.

1

Introduction and Main Result

Compressible Navier-Stokes-Poisson (N-S-P) equations with density-dependent viscosity coefficients in three-dimensional torus T3 can be described as: ⎧ ⎨ ∂t ρ + div(ρu) = 0, ∂t (ρu) + div(ρu ⊗ u) + (1.1) ´ ∇P (ρ) − div(ρDu) = ρ∇Φ, ⎩ λΦ = 4πG(ρ − |T13 | T3 ρdx), λ = ±1, with the initial conditions ρ(0, x) = ρ0 , ρu(0, x) = m0 ,

(1.2)

where t ≥ 0, x ∈ T3 ⊆ R3 , ρ = ρ(t, x) and u = u(t, x) represent the fluid density and T , and the pressure P velocity, respectively. Du is the strain tensor with Du = ∇u+∇u 2

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 2 is a non-monotone function of the density (see [12] for motivations) which satisfies the following conditions:  P ∈ C 1 (R+ ), P (0) = 0, (1.3)  1 γ−1 − b ≤ P (z) ≤ az γ−1 + b f or all z ≥ 0, az for two constants a > 0 and b ≥ 0. The term on the right hand side of the second equation in (1.1) describes the internal force of gradient vector field produced by potential functions. Moreover, for simplicity, ´ ´ using the conservation of mass T3 ρdx = T3 ρ0 dx, the poisson equation (1.1)3 can be normalized as λΦ = 4πG(ρ − 1), G > 0 is a fixed constant. From a physical point of view, the meaning of the N-S-P system is determined by the sign of the parameter λ. When λ > 0, the potential force Φ represents the electrostatic potential which produces the electric field E = −∇Φ and equations (1.1) are used to describe the transportation of charged particles in electronic devices, and then ρ ≥ 0, u represent the charge density and velocity, respectively. When λ < 0, the potential force Φ denotes the gravitational force and the N-S-P system (1.1) is used in astrophysics to describe the motion of gaseous stars, and then ρ ≥ 0, u denote the density, velocity of a gaseous star, respectively. In this paper, we mainly focus on the N-S-P system with degenerate viscosity, for the case of constant viscosity coefficients, the readers can refer to[7, 8, 11, 12, 22, 35]. For the case of density-dependent viscosity coefficients, the problem is much more challenge due to the degeneration near the vacuum and hence the obtained results are limit. Ducomet et.al in [13] studied the global stability of the weak solutions to the Cauchy problem of the N-S-P equations with non-monotone pressure as γ > 43 . In [14] they also considered Cauchy problem for the N-S-P equations of spherically symmetric motions in R3 , including both constant viscosities and density-dependent viscosities, and proved the global stability of the weak solutions provided that γ > 1. Zlotinik in [36] studied the long time behave of the spherically symmetric weak solutions near a hard core by giving global-in time bounds for the solutions. However, up to our knowledge, there are no results on a complete proof of the global existence of the weak solutions to the N-S-P equations except the system with some special structures, like spherically symmetric initial data. For instance, Zhang-Fang in [34] obtained the global well-posedness of the spherically symmetric weak solutions to the N-S-P system with small perturbation and eliminating a hard core; Duan-Li in [10] considered the free boundary problem and proved the existence of global spherically symmetric weak solutions as 65 < γ ≤ 43 . Moreover, when λ > 0, the Quasi-neutral limit is also an interesting problem which attracts many mathematicians, for more details, the readers can refer to [6, 9, 23, 25, 32] and references therein. Particularly, if without the poisson term ρ∇Φ, the equations (1.1) will be reduced to the classical barotropic compressible Navier-Stokes equations, the problem of the global weak solutions to which attracted many mathematicians to work on it. When the viscosity coefficients are constants, the major breakthrough was made by P.L.Lions in [20], where he proved the global existence of the finite energy weak solutions as long as γ ≥ N3N +2 , N = 2, 3. Later, Feireisl-Novotn´ y -Petzeltov´ a [17] and Feireisl [15] extended this result to γ > N2 , N = 2, 3, and Jiang-Zhang [24] and Sun-Jiang-Guo [29] improved it to γ > 1 under some symmetric conditions. When the viscosity coefficients are density-dependent, the problem turns to be much more difficult for its strong nonlinearity, degeneration and no well-definition for the velocity in the vacuum. To deal with this problem, the first

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 3 attempt was made by Bresch-Desjardins-Lin [5] and Bresch-Desjardins [2], in which when the viscosity coefficients satisfied a special relation, they established a new entropy-type √ inequality for the density (B-D entropy) to give a higher regularity as ∇ ρ ∈ L∞ (0, T ; L2 ) so that the compactness analysis of pressure turned easy. Then when proving the global stability of the weak solutions, the main difficulty turns to the lack of the strong con√ vergence of ρu in L2 (0, T ; L2 ). To overcome this difficulty, one approach was given by Bresch et al. in [2] by adding some additional terms like the drag terms r1 |u|u or the Korteweg term κρ∇ρ, and Mellet-Vasseur in [26] gave another approach by establishing the Mellet-Vasseur type inequality for ρu2 in L2 (0, T ; L log L(Ω)). For the construction of the approximate system, Bresch-Desjardins gave some ideas in [3, 4]. After that, some progresses were made on related models, J¨ ungel in [21] considered the compressible quantum Navier-Stokes equations and proved the existence of global weak solutions by choosing ρϕ as the test function, Gisclon-Violet in [18] also considered this system and showed the global existence of weak solutions but in the classical definition of weak solution by the use of a singular pressure close to the vacuum, where they also remarked that this singular pressure could be replaced by the drag force r0 u and r1 |u|u, and Zatorska in [33] proved the global existence of weak solutions to the Cauchy problem for the equations governing flow of isothermal reactive mixture of compressible gases with the help of ”cold” pressure. Then inspired by [3, 18, 21] and [33], Vasseur-Yu first proved the global existence of the weak solutions to the quantum Navier-Stokes equations with dampings in [30], which could be viewed as the approximate equations to the original Navier-Stokes system, meanwhile, they also proved the Mellet-Vasseur type inequality for the weak solutions by constructing some smooth multipliers in [31]. Then in combination with [30] and [31], they gave a complete proof for the existence of global weak solutions to the compressible Navier-Stokes equations with density-dependent viscosity, which answered a long open problem proposed by Lions in [20]. It should be noted that Li-Xin in [19] gave an another approach by constructing a new approximate system which satisfies the elementary energy estimates, B-D entropy and Mellet-Vasseur type inequality at the same time and established the global existence of weak solutions to the Navier-Stokes equations either in a periodic domain Ω ⊆ RN or in the whole space RN , N = 2, 3. Inspired by [33], [30] and [31], in the present paper, we are devoted to giving a complete proof of the global existence of weak solutions to the compressible N-S-P equations with degenerate viscosities, and we will prove that the problem admits a global-in-time weak solution as 43 < γ < 3 for the case λ = −1, or 1 < γ < 3 for the case λ = 1. To the best of our knowledge, this is the first result giving a complete proof of the global weak solutions to the N-S-P system with density-dependent viscosity, and our result holds true for the non-monotone pressure, which contains the classical γ law (P (ρ) = ργ ) case. Hence, our results are much more general. Throughout this paper, compared with the case λ = 1, the case λ = −1 is more difficult (negative energy in the energy estimates and some other additional bad terms needed to be controlled), hence we only focus on this case, and after some small modifications, the method can be directly applied to the case λ = 1, so we omit the details.

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 4

1.1

Formulation of the weak solutions and main result

For the smooth solutions (ρ, u, Φ(ρ)), multiplying the momentum equation (1.1)2 by u and integrating by parts, we can deduce the following energy inequality ˆ ˆ Tˆ 1 2 1 2 (1.4) ( ρu + Π(ρ) − ρ|Du|2 ≤ E0 , |∇Φ| )dx + 8πG 0 T3 2 T3 where ˆ

ρ

Π(ρ) = ρ 1

P (s) ds, s2

ˆ E0 =

1 1 ( ρ0 u20 + Π(ρ0 ) − |∇Φ(ρ0 )|2 )dx. 2 8πG 3 T

To prove the stability of the weak solutions (ρ, u, Φ(ρ)) of (1.1), we also need the following B-D entropy and Mellet-Vasseur type inequality as follows: ˆ Tˆ ˆ Tˆ γ 1 4 ∇ρ 2 2 2 |∇ρ | dxdt + ρ|∇u|2 dxdt ρ(u + ) dx + 2 ρ aγ 0 T3 0 T3 2 T3 ˆ 1  √ ≤ ρ0 u20 + |∇ ρ0 |2 dx + C, T3 2 ˆ ˆ ρ(1 + |u|2 ) ln(1 + |u|2 ) ≤ C ρ0 (1 + |u0 |2 ) ln(1 + |u0 |2 ) + C

ˆ

(1.5)

(1.6)

where C is bounded by the initial energy. Thus the initial data should satisfy the follows √ ρ0 ∈ L1 ∩ Lγ (T3 ), ρ0 ≥ 0, ∇ ρ0 ∈ L2 (T3 ), m0 ∈ L1 (T3 ), m0 = 0 if ρ0 = 0,

|m0 |2 ∈ L1 (T3 ), ρ0

(1.7)

ρ0 (1 + |u0 |2 ) ln(1 + |u0 |2 ) ∈ L1 (T3 ) Definition 1.1. We will say (ρ, u, Φ(ρ)) are the finite energy weak solutions of the problem (1.1)-(1.2) if the followings are satisfied: 1. ρ, u belong to the classes ⎧ √ ∞ 1 γ 3 ∈ L∞ ((0, T ); L2 (T3 )), ⎨ ρ ∈ L ((0, T ); L ∩ L (T )), ρu γ √ ∇ ρ ∈ L∞ ((0, T ); L2 (T3 )), ∇ρ 2 ∈ L2 ((0, T ); L2 (T3 )), ⎩ √ ρ∇u ∈ L2 ((0, T ); L2 (T3 )), ρ(1 + |u|2 ) ln(1 + |u|2 ) ∈ L∞ (0, T ; L1 (T3 )).

(1.8)



2. The equations (1.1)1 -(1.1)2 hold in the sense of D ((0, T ) × T3 ),(1.1)3 holds a.e. for (t, x) ∈ ((0, T ) × T3 ), 

3. (1.2) holds on D (T3 ), 4. (1.4), (1.5) and (1.6) hold for almost every t ∈ [0, T ]. Before stating the main result, we give some notations: Notations: Throughout this paper, C denotes a generic positive constant which may depend on the initial data or some other constants but independent of the indexes

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 5 ε, μ, η, δ, r0 , r1 and κ, C(·) > 0 means the constant C particularly depends on the parameters in the bracket, and ˆ ˆ ˆ Tˆ ˆ Tˆ f= f= f dx, f dxdt, T3

0

0

T3

f Lp = f Lp (T3 ) , f Lp (W s,r ) = f Lp (0,T ;W s,r (T3 )) . Then we state our main result: Theorem 1.1. Let λ = −1, 43 < γ < 3 and the initial data satisfy (1.7), then for any time T , there exists a weak solution (ρ, u, Φ) to (1.1)-(1.2) in the sense of Definition 1.1. Remark 1.1. It should be noted that the condition γ < 3 is required when deriving the Mellet-Vasseur type inequality (see (A.31)), which was first proved by Mellet-Vasseur in [26], and they showed that this restriction could be removed by adding an additional condition on the pressure, for the details, the readers can refer to [26]. Remark 1.2. For the case that λ = 1, with some small modifications, the proof in this paper can be directly applied to this case, and we can also prove that the system (1.1) admits a global weak solution provided that 1 < γ < 3. So in this paper, we omit the details for simplicity. The rest paper is organized as follows: In section 2, we state some elementary compactness theorems which will be used frequently in the whole proof. In section 3-6, we will first prove the global existence of weak solutions to N-S-P equations with additional damping terms and quantum potential by using the Faedo-Galerkin approximation and weak convergence method, and this system can be viewed as an approximate system of the origenal N-S-P equations. In section 7, follow the line in [31], we will prove the MelletVasseur type inequality for the weak solutions. In section 8, we complete the proof of the main theorem by passing to the limits as r0 = r1 → 0.

2

Preliminaries

The following two Lemmas are two standard compactness results and will help us get the strong convergence of the solutions: Lemma 2.1. [1, 28](Aubin-Lions Lemma) Let X0 , X and X1 be three Banach spaces with X0 ⊆ X ⊆ X1 . Suppose that X0 is compactly embedded in X and that X is continuously embedded in X1 . For 1 ≤ p, q ≤ +∞, let W = {u ∈ Lp ([0, T ]; X0 ) | ∂t u ∈ Lq ([0, T ]; X1 )}. (i) If p < +∞, then the embedding of W into Lp ([0, T ]; X) is compact. (ii) If p = +∞ and q > 1, then the embedding of W into C([0, T ]; X) is compact. Lemma 2.2. (Egorov’s theorem about uniform convergence) Let fn → f a.e. in Ω, a bounded measurable set in Rn , with f finite a.e. Then for any ε > 0 there exists a measurable subset Ωε ⊂ Ω such that |Ω\Ωε | < ε and fn → f uniformly in Ωε , moreover, if fn → f a.e. in Ω,

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 6 fn ∈ Lp (Ω) and unif ormly bounded , f or any 1 < p ≤ +∞, then, we have fn → f strongly in Ls (Ω), f or any s ∈ [1, p).

3

Faedo-Galerkin approximation

In this section, we construct the approximate system to the original problem (1.1) by using the Faedo-Galerkin method, we proceed it similarly in [[15], Chapter. 7] and [21].

3.1

Approximate the mass equation

Let T > 0, then we define a finite-dimensional space Xn =span{e1 , ...en }, n ∈ N, where {ek } is an orthonormal basis of L2 (T3 ) which is also an orthogonal basis of H 1 (T3 ). Let (ρ0 , u0 ) ∈ C ∞ (T3 ) be some initial data satisfying ρ0 ≥ ν > 0 for x ∈ (T3 ) for some ν > 0, and let the velocity u ∈ C([0, T ]; Xn ) be given with the following norm u(x, t) =

n 

λi (t)ei (x), (t, x) ∈ [0, T ] × T3

i=1

Note that Xn is a finite-dimensional space, all the norms are equivalence on Xn , so u is bounded in C([0, T ]; C k (T3 )) for any k ∈ N and there exists a constant C > 0 depending on k such that (3.1) uC([0,T ];C k (T3 )) ≤ CuC([0,T ];L2 (T3 )) . Then we approximate the continuity equation as follows:  ∂t ρ + div(ρu) = ερ, ρ0 ∈ C ∞ (T3 ), ρ0 ≥ ν > 0,

(3.2)

By the well-posedness theory of the parabolic equations ([16], Lemma 3.1) and the bootstrap method, it’s easy to prove that the system (3.2) exists an unique classical solution ρ ∈ C 1 ([0, T ]; C 7 (T3 )) and 0 < ρe−

´T 0

divuL∞ dt

≤ ρ(x, t) ≤ ρe

´T 0

divuL∞ dt

, ∀x ∈ T3 , t ≥ 0,

(3.3)

provided that 0 < ρ ≤ ρ0 ≤ ρ. Next we will show that the solution of the equation (3.2) depends on the velocity u continuously. Let ρ1 , ρ2 be two solutions with the same initial data, which means ∂t ρ1 + div(ρ1 u1 ) = ερ1 , ∂t ρ2 + div(ρ2 u2 ) = ερ2 . Subtracting the above two equations, multiplying the result equation with −(ρ1 −ρ2 ) and integrating by parts with respect to x over T3 , we have sup ρ1 − ρ2 H 1 ≤ τ C(ρ0 , ε, u1 , u2 L1 ((0,τ );W 1,∞ ) )u1 − u2 H 1 ,

t∈[0,τ ]

moreover, for u ∈ C([0, T ]; Xn ) is a given vector field, similarly by using the bootstrap method and compactness analysis, we can prove ρ1 − ρ2 C([0,τ ];C 7 (T3 )) ≤ τ C(ρ0 , ε, u1 , u2 L1 ((0,τ );Xn ) )u1 − u2 C([0,τ ];Xn ) .

(3.4)

If we introduce the operator S : C([0, T ]; Xn ) → C([0, T ]; C 7 (T3 )) by S(u) = ρ, we have the following Proposition:

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 7 Proposition 3.1. If 0 < ρ ≤ ρ0 ≤ ρ, ρ0 ∈ C ∞ (T3 ), u ∈ C([0, T ]; Xn ), then there exists an operator S : C([0, T ]; Xn ) → C([0, T ]; C 7 (T3 )) satisfying • ρ = S(u) is an unique solution to the problem (3.2). • 0 < ρe−

´T 0

divuL∞ dt

≤ ρ(x, t) ≤ ρe

´T 0

divuL∞ dt

, ∀x ∈ T3 , t ≥ 0.

• S(u1 ) − S(u2 )C([0,τ ];C 7 (T3 )) ≤ τ C(ρ0 , ε, u1 L1 ((0,τ );Xn ) , u1 L1 ((0,T τ );Xn ) )u1 − u2 C([0,τ ];Xn ) , for any τ ∈ [0, T ] and u1 , u2 ∈ Mk = {u ∈ C([0, T ]; Xn ); uC([0,T ];Xn ) ≤ k, t ∈ [0, T ]}. Remark 3.1. The proposition 3.1 shows the operator S is also Lipschitz continuous for sufficient small time t.

3.2

Approximate the momentum equations

Next we wish to solve the momentum equation on the space Xn by using the FaedoGalerkin approximation method. To this end, for given ρ = S(u), we are looking for a approximate solution un ∈ C([0, T ]; Xn ) satisfying ˆ ˆ ˆ Tˆ ρun (T )ϕdx − m0 ϕdx + div(ρun ⊗ un ) + ∇P (ρ) T3

T3 2

−6

0

T3

− div(ρDun ) + μ un − η∇ρ + ε∇ρ · ∇un + r0 un + r1 ρ|un |2 un ˆ Tˆ √  ρ

3 − δρ∇ ρ − κρ∇( √ ) ϕdxdt = ρ∇Φϕdxdt, ρ 0 T3

(3.5)

for any test function ϕ ∈ Xn . The extra term μ2 un is not only necessary to extend the local solution obtained by the fixed point theorem to a global one at the Gerlakin level but 2 2 also to make sure ∂t ( ∇ρ ρ ) ∈ L ((0, T ); L ) so that it can be taken as a test function when we compute the B-D entropy at the next level, the extra terms η∇ρ−6 and δρ∇3 ρ are used to keep the density bounded, and bounded away from below with a positive constant for all the time, this enables us to take ∇ρ ρ as a test function to derive the B-D entropy, and the term r0 un is used to control the density near the vacuum, the damping term r1 ρ|un |2 un √  ρ and quantum term κρ∇( √ρ ) are used to derive the Mellet-Vasseur type inequality for √ the weak solutions, which implies that ρu is strong convergence in L2 (0, T ; L2 (T3 )). To solve (3.5), we follow the same arguments as in [15, 17, 21], and introduce the following operator, giving a function ρ ∈ L1 (T3 ) with ρ > ρ > 0: ˆ ∗ M[ρ] : Xn → Xn , < M[ρ]u, v >= ρu · vdx, u, v ∈ Xn . T3

Similarly in [17], it’s easy to check that the operator M[ρ] satisfies the following properties: • M[ρ]L(Xn ,Xn∗ ) ≤ C(n)ρL1 . • M[ρ]L(Xn ,Xn∗ ) ≥ inf x∈T3 ρ • If inf x∈T3 ρ ≥ ρ > 0, then the operator M[ρ] is invertible with M−1 [ρ]L(Xn∗ ,Xn ) ≤ ρ−1 , where L(Xn∗ , Xn ) is the set of bounded liner mappings from Xn∗ to Xn .

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 8 • M−1 [ρ] is Lipschitz continuous in the sense M−1 [ρ1 ] − M−1 [ρ2 ]L(Xn∗ ,Xn ) ≤ C(n, ρ)ρ1 − ρ2 L1 (T3 ) for all ρ1 , ρ2 ∈ L1 (T3 ) such that ρ1 , ρ2 ≥ ρ > 0. Proof. Here, we omit the proof, for more details, the readers can refer to [15, 17, 21]. Then by using the operators M[ρ] and ρ = S(un ), we rewrite (3.5) as the following fixed-point problem un (t) = M

−1



[(S(un )(t)] M[ρ0 ](u0 ) +

where

ˆ 0

T

 N (S(un ), un )(s)ds ,

(3.6)

ˆ

< M[ρ0 ](u0 ), ϕ >Xn∗ ×Xn =

m0 ϕdx, ˆ √  ρ ρ∇Φ + κρ∇( √ ) + δρ∇3 ρ − div(ρun ⊗ un ) < N (S(un ), un )(s), ϕ >Xn∗ ×Xn = ρ

+ div(ρDun ) − μ2 un − ε∇ρ · ∇un − ∇P (ρ) + η∇ρ−6 − r0 un − r1 ρ|un |2 un ϕdx.

Thanks to the Lipschitz continuous estimates for S and M−1 , this equation can be solved   by using the fixed-point theorem of Banach for a short time [0, T ], where T ≤ T , in the space C([0, T ]; Xn ). Thus there exists a unique local-in-time solution (ρn , un , Φ(ρn )) to (3.2) and (3.5). Next we will extend this obtained local solution to be a global one. Differentiating (3.5) with respect to time t, taking ϕ = un and integrating by parts with respect to x over T3 , we have the following energy estimates ˆ ˆ ˆ ˆ 1d 1 2 2 ρun + ( ρt + div(ρun ))|un | + ρun · ∇un : un + ∇P (ρ) · un 2 dt 2 ˆ ˆ ˆ ˆ ˆ −6 2 2 − η ∇ρ · un + ε ∇ρ · ∇un · un + ρ|Dun | + r0 |un | + r1 ρ|un |4 (3.7) ˆ ˆ ˆ ˆ √  ρ + δ div(ρun )3 ρ + μ |un |2 = − div(ρun )Φ − κ div(ρun ) √ . ρ First, using the integration by parts and the equation (3.2)1 , we have ˆ ˆ ˆ 1 2 2 ρt |un | + div(ρu)|un | + ρun · ∇un : un 2 ˆ ˆ ˆ (3.8) 1 2 i j j = [ (ερ − div(ρun )) + div(ρu)]|un | + ρun ∂i un un = −ε ∇ρ · ∇un · un , 2 and ˆ

ˆ

ˆ ˆ ρ   P (s) P (s) ds (ρun )dx = − ds[ερ − ρt ]dx s s 1 1 ˆ ˆ  P (ρ) d Π(ρ)dx + ε = |∇ρ|2 , dt ρ

∇P (ρ) · un =

ˆ

∇

ρ

(3.9)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 9 ´ρ where Π(ρ) = ρ 1 Ps(s) Next we will deal with the cold pressure and high order 2 ds. derivative of the density terms: ˆ ˆ ˆ ˆ 6 −6 3 −7 −η ∇ρ · un + δ div(ρun ) ρ = − η ρun · ∇ρ + δ [ερ − ρt ]3 ρ 7 ˆ ˆ ˆ ˆ (3.10) 2 δ d 1 d −6 −3 2 2 ρ + ηε |∇ρ | + |∇ρ| + δε |2 ρ|2 . = η 7 dt 3 2 dt The terms on the right hand side can be controlled as follows: ˆ ˆ ˆ ˆ − div(ρun )Φ = − [ερ − ρt ]Φ = ∂t (ρ − 1)Φ − ε (ρ − 1)Φ ˆ ˆ 1 d 2 = |∇Φ| + ε 4πG(ρ − 1)2 , 8πG dt ˆ

and −κ

(3.11)

ˆ ˆ √  ρ κε √ 2 2 ρ|∇ log ρ| − κ ∂t |∇ ρ|2 , div(ρun ) √ = − ρ 2

(3.12)

√  ρ

where we used the identity ρ∇( √ρ ) = 12 div(ρ∇2 log ρ) and equation (1.1)3 . Then substituting (3.8)-(3.12) into (3.7) and integrating the result equation with respect to t over [0, T ], yields ˆ

ˆ

P  (ρ) 2 |∇ρ|2 + ηε|∇ρ−3 |2 + rho|Dun |2 + r0 |un |2 + r1 ρ|un |4 E(t) + ε ρ 3 0 ˆ Tˆ κε (ρ − 1)2 dxdt + E0 , + μ|un |2 + δε|2 ρ|2 + ρ|∇2 log ρ|2 ]dxdt = 4πGε 2 0 T

(3.13)

where ˆ

1 η |∇Φ(ρ)|2 δ √ E(t) = ( ρu2n +Π(ρ)+ ρ−6 − + |∇ρ|2 +κ|∇ ρ|2 )dx, Π(ρ) = ρ 7 8πG 2 T3 2 and

ˆ

ρ 1

P (s) ds s2

ˆ E0 =

1 η 1 δ √ ( ρ0 u2n + Π(ρ0 ) + ρ−6 |∇Φ(ρ0 )|2 + |∇ρ0 |2 + κ|∇ ρ0 |2 )dx. 0 − 7 8πG 2 T3 2

Moreover, because of (1.3), we have ˆ

ρ

Π(ρ) = ρ 1

P (s) ds ≥ ρ s2

and ˆ Tˆ ˆ Tˆ  P (ρ) 2 ε |∇ρ| ≥ ε ρ 0 0

ˆ

ρ 1 sγ aγ

− bs

1

s2

1 γ−1 aρ

−b

ρ

ds =

1 (ργ − ρ) − bρ log ρ, aγ(γ − 1)

4ε |∇ρ| = 2 aγ

ˆ

2

T

ˆ

γ 2

ˆ

|∇ρ | − bε

0

2

T

0

ˆ

(3.14)

1 |∇ρ|2 , ρ (3.15)

furthermore, if γ > 43 , by the standard elliptic estimate, we have 1 8πG

ˆ

5γ−6

γ

|∇Φ|2 ≤ Cρ − 12 6 ≤ C(1 + ρL3(γ−1) ρL3(γ−1) ) ≤ C + ςργLγ , γ 1 L5

(3.16)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 10 where 0 < ς  1 is a fixed constant. Then substituting (3.14)-(3.16) into (3.13), we have ˆ ˆ Tˆ 4ε γ 1 2 ργ η −6 δ √ 2 2 |∇ρ 2 |2 ρun + + ρ + |∇ρ| + κ|∇ ρ| dx + 2 2 aγ(γ − 1) 7 2 aγ 0

2ηε κε + |∇ρ−3 |2 + ρ|Dun |2 + r0 |un |2 + r1 ρ|un |4 + μ|un |2 + δε|2 ρ|2 + ρ|∇2 log ρ|2 3 2 ˆ ˆ Tˆ 1 1 (ρ − 1)2 dxdt + [ ≤ 4πGε ρ + bρ log ρ + bε |∇ρ|2 ]dx + C + ςργLγ αγ(γ − 1) ρ 0 ˆ Tˆ ˆ 1  ρ2 + bε ≤ ς ργLγ + C + Cε |∇ρ|2 , ρ 0 (3.17)  where ς is a sufficient small positive constant, C is a generic positive constant only depending on the initial data and T . Since ˆ T ˆ Tˆ ˆ 4 2 ε T 2 3 2 3 3 (3.18) ρL1 ∇ ρL2 + ρL1 dt ≤ C + C δ∇3 ρ2L2 dt Cε ρ ≤ Cε δ 0 0 0 and

ˆ bε 0

T

ˆ

1 |∇ρ|2 ≤ Cε ρ

ˆ

T

7

11

ρ−1 L6 (ρL9 1 ∇3 ρL92 + ρ2L1 )dt 0 ˆ ˆ Cε T Cε T −1 6 ηρ L6 dt + δ∇3 ρ2L2 dt, ≤C+ η 0 δ 0

(3.19)

then substituting (3.18)-(3.19) into (3.17) and using the Gronwall inequality gives ˆ ε ε C( ε + ε )T ηρ−6 + δ|∇ρ|2 dx ≤ C[1 + ( + )T e δ η ]. (3.20) δ η In combination with (3.17) and (3.20), we have the following energy inequality ˆ ˆ Tˆ 4ε γ 1 2 1 η −6 δ √ 2 γ 2 |∇ρ 2 |2 ρun + ρ + ρ + |∇ρ| + κ|∇ ρ| dx + 2 2 aγ(γ − 1) 7 2 aγ 0 2 (3.21) + ηε|∇ρ−3 |2 + ρ|Dun |2 + r0 |un |2 + r1 ρ|un |4 + μ|un |2 + δε|2 ρ|2 3 κε ε ε ε ε C( ε + ε )T + ρ|∇2 log ρ|2 ]dxdt ≤ C + C( + )[1 + ( + )T e δ η ]T. 2 δ η δ η So the energy inequality (3.21) yields ˆ T un 2L2 dt ≤ C(ε, η, δ) < +∞. 0

Due to dimXn < ∞ and (3.3), then the density is bounded and bounded away from blow with a positive constant, which means there exists a constant c > 0 such that 0<

1 ≤ ρn ≤ c, c

for all t ∈ [0, T ∗ ). Furthermore, the energy inequality also gives us ˆ sup ρn u2n ≤ C < ∞, t∈(0,T ∗ )

(3.22)

(3.23)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 11 then together with (3.22) and (3.23), implies supt∈(0,T ∗ ) un L∞ ≤ C < ∞, where we used the fact that all the norms are equivalence on Xn . Then we can repeat above argument many times and use the compactness analysis, we can obtain un ∈ C([0, T ]; Xn ), then we can extend T ∗ to T . Thus there exists a global solution (ρn , un , Φ(ρn )) to (3.2), (3.5) for any time T with satisfying the energy estimates (3.21). Before we continue the further analysis, we need the following Lemma: Lemma 3.2. For any smooth positive function ρ(x), we have ˆ ˆ ˆ 1 √ ρ|∇2 log ρ|2 dx ≥ c1 |∇2 ρ|2 dx + c2 |∇ρ 4 |4 dx, where c1 and c2 are two positive constants. Remark 3.2. This inequality was first proved by J¨ ungel [21], and later Vasseur-Yu [31] gave a quick proof. For the details, the readers can refer to these two papers.

3.3

Passing to the limits as n → ∞.

Then we pass to the limits as n → ∞, with ε, μ, η, δ, r0 , r1 , κ being fixed, this objective can be achieved through the following steps: 3.3.1

Step1.Convergence of ρn , Pressure P (ρn ) and gravitational force ∇Φ(ρn ).

By (3.2), we have ˆ 0

T

ˆ

ˆ (ρn )t ϕ = −ε

T

ˆ

0

ˆ ∇ρn ∇ϕ +

T

ˆ

0

(ρn un )∇ϕ

≤ (C∇ρn L2 (0,T ;L2 ) + Cρn L∞ (0,T ;L∞ ) un L2 (0,T ;L2 ) )∇ϕL2 (0,T ;L2 ) ≤ C, holds for any ϕ ∈ L2 (0, T ; H 1 ), which yields ∂t ρn ∈ L2 (0, T ; H −1 ). This together with ρn ∈ L∞ (0, T ; H 3 ) ∩ L2 (0, T ; H 4 ), using the Aubin-Lions Lemma, we can claim ρn ∈ C([0, T ]; H 3 ), so up to a subsequence, we have ρn → ρ strongly

in C([0, T ]; H 3 ), hence, ρn → ρ a.e. 5 3

(3.24)

5 3

Next we claim that ργn is bounded in L (0, T ; L ). γ

Notice that ∇ρn2 is bounded in L2 (0, T ; L2 ), using the Sobolev embedding theorem older inequality to have gives us ργn is bounded in L1 (0, T ; L3 ), then we apply H¨ ργn 

5 5 L 3 (0,T ;L 3 )

2

3

≤ ργn L5 ∞ (0,T ;L1 ) ργn L5 1 (0,T ;L3 ) ≤ C. 5

5

Similarly, we can show ρ−6 n is bounded in L 3 (0, T ; L 3 ) too. Moreover, for ρn → ρ a.e.,  γ γ so ρn → ρ a.e.. Recall that the pressure satisfies a1 ργ−1 − b ≤ P ≤ aργ−1 + b and P ∈ C 1 (R+ ), integrating this inequality we have |P (ρn )| ≤ C(ργn + ρn ), it implies that 5 5 5 5 P (ρn ) is bounded in L 3 (0, T ; L 3 ) due to ργn is bounded in L 3 (0, T ; L 3 ). For ργn → ργ a.e., using the Egorov’s theorem, we have P (ρn ) → P (ρ)

strongly

in L1 (0, T ; L1 ).

(3.25)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 12 Next, we show that the density is bounded away from zero with a positive constant for all the time t ∈ [0, T ] by using the Sobolve inequality. For 1

1

−1 2 2 −1 2 −1 ρ−1 n L∞ ≤ C(ρn L6 ∇ ρn L2 + ρ L6 ),

(3.26)

−2 2 −3 2 ∇2 ρ−1 n L2 ≤ C(ρn ∇ ρn L2 + ρn (∇ρn ) L2 ) 2 3 −1 3 2 ≤ Cρ−1 n L6 (ρn L1 + ∇ ρn L2 ) + Cρn L6 ∇ρn L∞

≤ C(1 +

3 ρ−1 n L6 )(1

+ ∇

3

(3.27)

ρn 2L2 ),

substituting (3.27) into (3.26), yields 1

3

−1 2 −1 3 2 ρ−1 n L∞ ≤ Cρn L6 (1 + ρn L6 ) (1 + ∇ ρn L2 )

2 3 ≤ C(1 + ρ−1 n L6 ) (1 + ∇ ρn L2 ) ≤ C(η, δ, T ),

(3.28)

where here the constant C(η, δ, T ) depends on η, δ and T but independent of n. −6 a.e., together So immediately, we have ρ1n → ρ1 a.e., furthermore we show ρ−6 n → ρ 5

5

with ρ−6 n ∈ L 3 (0, T ; L 3 ) and Egorov’ theorem, we have −6 ρ−6 n →ρ ,

strongly

in L1 (0, T ; L1 ).

In combination with (3.16) and ρn convergence to ρ strongly in C([0, T ]; H 3 ), we have ∇Φ(ρn ) → ∇Φ(ρ) strongly in L2 (0, T ; L2 ). 3.3.2

Step2. Convergence of momentum ρn un

From the energy estimates, we know that un is bounded in L2 (0, T ; L2 ), so up to a subsequence, we have un  u in L2 (0, T ; L2 ), recall that ρn → ρ strongly in C([0, T ]; H 3 ), so we have ρn un  ρu weakly in L1 (0, T ; L1 ). Moreover, since ρn ∈ L∞ (0, T ; H 3 ), un ∈ L2 (0, T ; H 2 ), we have ∇(ρn un ) = ∇ρn un + ρn ∇un ∈ L2 (0, T ; L2 ), together with ρn un ∈ L2 (0, T ; L2 ), we have ρn un ∈ L2 (0, T ; H 1 ). Next in order to use the Aubin-Lions Lemma, we only need to prove ∂t (ρn un ) ∈ L2 (0, T ; H −s ), for some s > 0. Since, 2 ∂t (ρn un ) = −div(ρn un ⊗ un ) − ∇P (ρn ) + η∇ρ−6 n − μ un + div(ρn Dun ) √  ρn 2 3 ), − r0 un − r1 ρn |un | un − ε∇ρn · ∇un + δρn ∇ ρn + ρn ∇Φ + κρn ∇( √ ρn

(3.29)

due to the energy estimates (3.21), it’ s easy to check that ∂t (ρn un ) ∈ L2 (0, T ; H −3 ), then by using the Aubin-Lions Lemma, we can show ρn un → g strongly in L2 (0, T ; L2 ), f or some f unction g ∈ L2 (0, T ; L2 ), furthermore, due to ρn un  ρu weakly in L1 (0, T ; L1 ), we have ρn un → ρu strongly in L2 (0, T ; L2 ).

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 13 3.3.3

Step.3 Convergence of nonlinear diffusion terms.

Let ϕ ∈ C ∞ ([0, T ]; T3 ), using integration by parts, we have ˆ Tˆ ˆ Tˆ ∂i ujn + ∂j uin div(ρn Dun )ϕ = ∂i (ρn ( ))ϕ 2 0 0 ˆ ˆ ˆ ˆ 1 T 1 T j j [(ρn un )∂ii ϕ + ∂i ρn un ∂i ϕ] + [(ρn uin )∂ij ϕ + ∂j ρn uin ∂i ϕ], = 2 0 2 0

(3.30)

since ρn → ρ strongly in C([0, T ]; H 3 ), ρn un → ρu strongly in L2 (0, T ; L2 ), un  u weakly in L2 (0, T ; L2 ), we have ˆ Tˆ ˆ Tˆ ˆ Tˆ ˆ Tˆ j j j (ρn un )∂ii ϕ → (ρu )∂ii ϕ, ∂i ρn un ∂i ϕ → ∂i ρuj ∂i ϕ, ˆ

0

T

0

ˆ

ˆ (ρn uin )∂ij ϕ →

0

T

0

ˆ

ˆ (ρui )∂ij ϕ,

T

0

ˆ

ˆ ∂j ρn uin ∂i ϕ →

T

ˆ ∂j ρui ∂i ϕ.

0 0 0 ´T ´ 3 2 And for 0 ρn ∇ ρn ϕ = − 0 (ρn divϕ + ϕ · ∇ρn ) ρn , we only focus on the ´T ´ most difficult term − 0 ϕ · ∇ρn 2 ρn , as ρn → ρ strongly in C([0, T ]; H 3 ) and ρn  ρ ´T ´ ´T ´ in L2 (0, T ; H 4 ), we can obtain − 0 ϕ · ∇ρn 2 ρn → − 0 ϕ · ∇ρ2 ρ. ´T ´ Then we apply the above arguments to handle the other terms from − 0 div(ρn ϕ)3 ρn , ´T ´ ´T ´ we have 0 ρn ∇3 ρn ϕ → 0 ρ∇3 ρϕ, as n → ∞. √ √ Similar as (3.24), we can also obtain ∇ ρn → ∇ ρ strongly in L2 (0, T ; L2 ) and √ √  ρn   ρ in L2 (0, T ; L2 ), which implies

´T ´

ˆ Tˆ ˆ Tˆ √  ρn √ √ √ √ ρn ∇( √ ϕ∇ ρn  ρn )ϕ = −κ ρn  ρn divϕ − 2κ κ ρn 0 0 0 ˆ Tˆ ˆ Tˆ √ √ √ √ ϕ∇ ρ ρ. → −κ ρ ρdivϕ − 2κ ˆ

T

ˆ

0

0

With the above compactness results in hand, we are ready to pass to the limits as n → ∞ in the approximate system (3.2), (3.5). Thus, we can show that (ρ, u, Φ) solves ρt + div(ρu) = ερ, a.e. on (0, T ) × T3 , Φ = −4πG(ρ − 1), a.e. on (0, T ) × T3 , (ρu)t + div(ρu ⊗ u) + ∇P (ρ) − η∇ρ−6 − div(ρDu) + μ2 u + ε∇ρ · ∇u + r0 u + r1 ρ|u|2 u √  ρ − δρ∇3 ρ − κρ∇( √ ) = ρ∇Φ, holds in the sense of distribution on (0, T ) × T3 . ρ (3.31) Thanks to the weak lower semi-continuity of norms, we pass to the limits in the energy estimate (3.21), yields ˆ Tˆ 4ε γ 2 |∇ρ 2 |2 + ηε|∇ρ−3 |2 + ρ|Du|2 + r0 |u|2 + r1 ρ|u|4 sup E(t) + 2 aγ 3 0 t∈(0,T ) (3.32)

κε 2 2 2 2 2 + μ|u| + δε| ρ| + ρ|∇ log ρ| dxdt ≤ C + Cε , 2 ´ 1 2 √ η 1 ργ + 7 ρ−6 + 2δ |∇ρ|2 + κ|∇ ρ|2 )dx, and Cε = C( δε + ηε )[1 + where E(t) = ( 2 ρu + aγ(γ−1) ( δε + ηε )T e

C( δε + ηε )T

]T.

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 14

B-D entropy and passing to the limits as ε, μ → 0

4

In this section, we are going to deduce the B-D entropy estimate which was first established by Bresch-Desjardins-Lin in [5]. By (3.28),(3.32) and u ∈ L2 (0, T ; H 2 ), we have ρ(x, t) ≥ C(δ, η) > 0, ρ ∈ L∞ (0, T ; H 3 ) ∩ L2 (0, T ; H 4 ), and ∂t ρ ∈ L2 (0, T ; L2 ). (4.1)

4.1

B-D entropy.

Thanks to (4.1), it’s easy to check can use

∇ρ ρ

∇ρ ρ

2 2 ∈ L2 (0, T ; H 3 ) and ∂t ∇ρ ρ ∈ L (0, T ; L ), so we

as a test function to test the momentum equation to derive the B-D entropy: ˆ ˆ ˆ d ∇ρ 2 1 ∇ρ 2 2 2 −3 2 ( ρ| | + ρu )dx + δ | ρ| + η |∇ρ | − ρ∂i uj ∂j ui dt 2 ρ ρ 3 ˆ ˆ Tˆ 1 κ ρ|∇2 log ρ|2 +ε |ρ|2 + ρ 2 0 ˆ ˆ (4.2)

∇ρ ∇ρ = ρ∇Φ − ∇P (ρ) − r0 u − r1 ρ|u|2 u · − μ u · ( ) ρ ρ ˆ ˆ ˆ 5 ∂j ρ ε ∇ρ 2  ρ ρ| Ii . − ε div(ρu) − ε ∂i ρ∂i uj + | = ρ ρ 2 ρ ˆ

i=1

By the equation (1.1)3 and the condition (1.3),we have ˆ  P Φ(ρ − 1) − |∇ρ|2 − r0 ρ−1 div(ρu) + r1 ρ(|u|2 divu + 2ui uk ∂i uk )] I1 = − ρ ˆ ˆ 1 γ−1 ˆ −b 2 2 aρ |∇ρ| + r0 ∂t log ρ ≤ C (ρ − 1) − ρ ˆ 1√ 1 √ − r0 ε |∇ρ|2 + Cr12 ρu2 L2  ρ∇uL2 2 ρ ˆ ˆ 1√ γ 4 √ 2 √ 2 2 ≤ (C(ρ − 1) − 2 |∇ρ 2 | + 4b|∇ ρ| )dx + r0 ∂t log ρ + Cr12 ρu2 L2  ρ∇uL2 aγ (4.3) Substituting (4.3) into (4.2) and integrating it with respect to the time t over [0,T], we have ˆ Tˆ ˆ 2 γ 1 4 ∇ρ 2 ρ(u + ) − r0 log ρ dx + η|∇ρ−3 |2 + ρ|∇u|2 + 2 |∇ρ 2 |2 + δ|2 ρ|2 2 ρ 3 aγ 0

κ 2 2 + ρ|∇ log ρ| dxdt 2 ˆ Tˆ ˆ

1 1 √ ρ|Du|2 + 4b|∇ ρ|2 + C(ρ − 1)2 + Cr1 ρ|u|4 + ρ|∇u|2 dxdt ρ|u|2 dx + ≤ 2 2 0 ˆ T ˆ 5 1 ∇ρ0 2 Ii dt + ) − r0 log ρ0 dx + ρ0 (u0 + 2 ρ0 0 i=2 ˆ Tˆ ˆ T ˆ 5 1 T √ √ 4b|∇ ρ|2 + C(ρ − 1)2 dxdt +  ρ∇u2L2 dt + Ii dt, ≤ C + Cε + 2 0 0 0 i=2 (4.4)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 15 where we used the energy inequality (3.32). Moreover, to make the right hand side of (4.4) is bounded, we also need to require − log ρ0 ∈ L1 (T3 ), and this condition could be removed at the end of paper. Now we need to control the rest terms on the right hand of the (4.4): ˆ

T

ˆ

ˆ

5γ−6

T

5γ

(ρ − 1) ≤ C + C (ρL2(5γ−3) ρ 2(5γ−3) )2 dt 1 5γ L 3 0 0 (4.5) ˆ T ˆ T 5γ−6 2 3  5  γ 2 (5γ−3) γ 5 γ 5 5γ−3 2 3 ρL1 dt ≤ 2 ∇ρ 2 L2 dt + (C + Cε ) , ρ L1 ρ L3 ≤C +C aγ 0 0 2

C

6 5γ−3

where we require ˆ

ˆ

T

T

ˆ

< 2, which implies γ > 65 .

ˆ

1 ∇ρ 1 ∇ρ

I2 dt = −μ u div∇2 ρ − 2 ∇2 ρ − 2 div(∇ρ ⊗ ∇ρ) + 2∇ρ ⊗ ∇ρ 3 ρ ρ ρ ρ 0 0 ˆ Tˆ 1 3

1 1 ≤ Cμ |u| |∇ ρ| + 2 |∇ρ||∇2 ρ| + 3 |∇ρ|3 ρ ρ ρ 0 1 1 √ √ ≤ C μ μuL2 (L2 )  L∞ (L∞ ) ∇3 ρL2 (L2 ) +  2L∞ (L∞ ) ∇ρL∞ (L∞ ) ∇2 ρL2 (L2 ) ρ ρ

1 +  3L∞ (L∞ ) ∇ρ3L6 (L6 ) ρ 1 √ √ √ ≤ C μ μuL2 (L2 ) ( 3L∞ (L∞ ) + 1)(∇3 ρ3L∞ (L2 ) + 1) ≤ C(δ, η)(C + Cε )s μ, ρ (4.6)

ˆ

T

T

0

ˆ I3 dt = −ε

ˆ

ˆ

u · ∇ρ ρ ρ 0 + CεuL2 (L2 ) ρ−1 L∞ (L∞ ) ∇ρL∞ (L∞ ) ρL2 (L2 )

divuρ − ε

0

T

≤ Cε∇uL2 (L6 ) ρ 2 65 L (L ) √ √ 2 ≤ C ε( ε∇ uL2 (L2 ) + uL2 (L2 ) )(ρL∞ (L1 ) + ∇3 ρL∞ (L2 ) )

√ + CεuL2 (L2 ) ρ−1 L∞ (L∞ ) (ρL∞ (L1 ) + ∇3 ρL∞ (L2 ) )2 ≤ C(δ, η, r0 )(C + Cε )s ( ε + ε), (4.7) ˆ

√ √ I4 dt ≤ C ερ−1 L∞ (L∞ ) ∇ρ2L4 (L4 )  ε∇uL2 (L2 ) 0 √ √ ≤ C(δ, η) ερ−1 L∞ (L∞ ) (∇3 ρL∞ (L2 ) + ρL∞ (L1 ) )2 ( ε∇2 uL2 (L2 ) + uL2 (L2 ) ) √ ≤ C(δ, η, r0 )(C + Cε )s ε, (4.8) T

and ˆ 0

T

ˆ I5 dt ≤ Cε ≤

T

ρ−1 2L∞ ρL2 ∇ρ2L4 dt

0 Cερ−1 2L∞ (L∞ ) (∇3 ρL∞ (L2 )

where s > 0 is a suitable large constant.

(4.9)

+ ρL∞ (L1 ) ) ≤ C(δ, η)(C + Cε ) ε, 3

s

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 16 Then substituting (4.5)-(4.9) into (4.4), we have ˆ ˆ Tˆ ˆ ˆ 1 2 1 T ∇ρ 2 −3 2 ρ(u + ) − r0 log ρdx + η |∇ρ | + ρ|∇u|2 2 ρ 3 0 2 0 ˆ Tˆ ˆ Tˆ ˆ ˆ γ 2 κ T 2 2 2 2 |∇ρ | + δ | ρ| + ρ|∇2 log ρ|2 + 2 aγ 0 2 0 0 ˆ Tˆ √ √ √ |∇ ρ|2 + C(δ, η, r0 )(C + Cε )s (ε + ε + μ) + (C + Cε )3 , ≤ 4b

(4.10)

0

by using the Gronwall inequality, yields ˆ √ √ √ |∇ ρ|2 dx ≤ [C(δ, η, r0 )(C + Cε )s (ε + ε + μ) + (C + Cε )3 ](1 + 4bT e4bT ), so with this inequality and (4.10), we have the following B-D entropy estimats ˆ ˆ ˆ ˆ Tˆ ∇ρ 2 1 2 1 T −3 2 ρ(u + |∇ρ | + ρ|∇u|2 ) − r0 log ρdx + η 2 ρ 3 0 2 0 ˆ Tˆ ˆ Tˆ ˆ ˆ γ 2 κ T 2 2 2 2 |∇ρ | + δ | ρ| + ρ|∇2 log ρ|2 + 2 aγ 0 2 0 0 √ √ s ≤ [C(δ, η, r0 )(C + Cε ) (ε + ε + μ) + (C + Cε )3 ][(1 + 4bT e4bT )T + 1], ε

(4.11)

(4.12)

ε

where Cε = C( δε + ηε )[1 + ( δε + ηε )T eC( δ + η )T ]T , and s > 0 is a suitable large fixed constant.

4.2

Passing to the limits as μ, ε → 0.

We use (ρμ,ε , uμ,ε , Φ(ρμ,ε )) to denote the solutions at this level of approximation. From the energy estimates (3.32) and B-D entropy (4.12), we have the following compactness results: Lemma 4.1. Let (ρμ,ε , uμ,ε , Φ(ρμ,ε )) be weak solutions to (3.31), we have ρμ,ε ∈ L2 (0, T ; H 4 ), ∂t ρμ,ε ∈ L2 (0, T ; H −1 ), 3

ρμ,ε uμ,ε ∈ L2 (0, T ; W 1, 2 ), ∂t (ρμ,ε uμ,ε ) ∈ L2 (0, T ; H −3 ), 5

5

5

5

3 3 ργμ,ε ∈ L 3 (0, T ; L 3 ), ρ−6 μ,ε ∈ L (0, T ; L ), 5

(4.13)

5

P (ρμ,ε ) ∈ L 3 (0, T ; L 3 ), Φ(ρμ,ε ) ∈ L∞ (0, T ; H 2 ), by using Aubin-Lions Lemma, we can obtain the following compactness results ρμ,ε → ρ, a.e. and strongly in C([0, T ]; H 3 ), ρμ,ε  ρ, in L2 (0, T ; H 4 ), P (ρμ,ε ) → P (ρ), a.e. and strongly in L1 (0, T ; L1 ), uμ,ε  u, in L2 (0, T ; L2 ), ρμ,ε uμ,ε → ρu strongly in L2 (0, T ; Lp ), f or ∀1 ≤ p < 3, −6 1 1 ρ−6 μ,ε → ρ , a.e. and strongly in L (0, T ; L ),

∇Φ(ρμ,ε ) → ∇Φ(ρ) strongly in L2 (0, T ; L2 ), − ρμ,ε Duμ,ε → −ρDu, in the sense of distribution on (0, T ) × T3 , − δρμ,ε ∇3 ρμ,ε → −δρ∇3 ρ, in the sense of distribution on (0, T ) × T3 , √ √  ρμ,ε  ρ − κρμ,ε ∇( √ ) → −κρ∇( √ ), in the sense of distribution on (0, T ) × T3 . ρμ,ε ρ

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 17 Proof. The proof is similar as that in section 3, we repeat the compactness arguments here again and for the simplicity, we omit the details. With the above compactness results in hand, then we pass to the limits as μ = ε → 0, here we only focus on the terms involving with ε and μ. Firstly, since ρμ,ε is bounded in L∞ (H 3 ) ∩ L2 (H 4 ) uniformly on ε, we have ˆ

T

ε 0

ˆ ρμ,ε ϕ ≤ ερμ,ε L2 (L2 ) ϕL2 (L2 ) → 0, as ε → 0,

similarly, ˆ Tˆ √ √ ε ∇ρμ,ε · ∇uμ,ε ϕ ≤ ε∇ρμ,ε L2 (L2 )  ε∇uμ,ε L2 (L2 ) ϕL∞ (L∞ ) → 0, as ε → 0, 0

and

ˆ

T

μ 0

ˆ 2 uμ,ε ϕ ≤

√ √ μ μuμ,ε L2 (L2 ) ϕL2 (L2 ) → 0, as μ → 0.

So passing to the limits as μ = ε → 0 in (3.31), we have ρt + div(ρu) = 0, a.e. on (0, T ) × T3 (ρu)t + div(ρu ⊗ u) + ∇P (ρ) − η∇ρ−6 − div(ρDu) + r0 u + r1 ρ|u|2 u − δρ∇3 ρ √  ρ − κρ∇( √ ) = ρ∇Φ, holds in the sense of distribution on (0, T ) × T3 , ρ

(4.14)

Φ = −4πG(ρ − 1), holds a.e. on (0, T ) × T3 . Furthermore, thanks to the weak lower semi-continuity of the convex function and the strong convergence of ρμ,ε , uμ,ε , Φ(ρμ,ε ), we can pass to the limits in the energy inequality (3.32) and Bresch-Desjardins entropy (4.12) as μ = ε → 0 with δ, η, r0 , r1 , κ being fixed, ˆ

ˆ Tˆ 1 η −6 δ 1 2 γ 2 ρ|Du|2 ρu + ρ + ρ + |∇ρ| dx + 2 aγ(γ − 1) 7 2 0 ˆ Tˆ ˆ Tˆ |u|2 + r1 ρ|u|4 ≤ C(T ), + r0 0

(4.15)

0

and ˆ

∇ρ 2 ρ(u + ) − r0 log ρdx + ρ ˆ Tˆ ˆ Tˆ γ 2 2 2 |∇ρ | + δ + 2 aγ 0 0

1 2

ˆ ˆ 1 T |∇ρ | + ρ|∇u|2 2 0 0 ˆ Tˆ κ |2 ρ|2 + ρ|∇2 log ρ|2 ≤ C(T ). 2 0

2 η 3

ˆ

T

ˆ

−3 2

where we used the fact that Cε = C( δε + ηε )[1 + ( δε + ηε )T e

5

C( δε + ηε )T

(4.16)

]T → 0, as ε → 0.

Passing to the limits as η → 0.

In this section, we pass to the limits as η → 0 with δ, r0 , r1 , κ being fixed. we denote that (ρη , uη , Φ(ρη )) are weak solutions at this level, due to the energy estimates (4.15) and the

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 18 B-D entropy (4.16), it’s easy to check that we have the same estimates as that in Lemma 4.1 at the level with η, thus we can deduce the same compactness for (ρη , uη , Φ(ρη )). So at this level of approximation, we only focus on the convergence of the term η∇ρ−6 η . Here we state the following Lemma. Lemma 5.1. For ρη satisfying (4.15), we have ˆ Tˆ ρ−6 η η dxdt → 0 0

as η → 0. Proof. The proof is almost exactly the same as that in Vasseur-Yu [31], here we state the details for completeness. From the B-D entropy (4.16), we have ˆ 1 (log( ))+ dx ≤ C(r0 ) < +∞. (5.1) sup ρη t∈[0,T ] Note that y ∈ R+ → log( y1 )+ is a convex continuous function. Moreover, in combination with the property of the convex function and Fatou’s Lemma, yields ˆ ˆ ˆ 1 1 1 lim inf(log( ))+ dx ≤ lim inf (log( ))+ dx, (log( ))+ dx ≤ (5.2) η→0 η→0 ρ ρη ρη which implies (log( ρ1 ))+ is bounded in L∞ (0, T ; L1 ), so it allows us to deduce that |{x | ρ(t, x) = 0}| = 0, f or almost every t ∈ [0, T ],

(5.3)

where |A| denotes the measure of set A. Thanks to the compactness of the density: ρη → ρ strongly in C([0, T ]; H 3 ), hence ρη → ρ a.e., then together with (5.3), we deduce ηρ−6 η → 0 a.e. Moreover,using the interpolation inequality, yields ηρ−6 η 

5 5 L 3 (0,T ;L 3 )

2

3

−6 5 5 ≤ ηρ−6 η L∞ (0,T ;L1 ) ηρη L1 (0,T ;L3 ) ≤ C,

1 1 then using the Eogrov’s theorem, we have ηρ−6 η → 0, strongly in L (0, T ; L ).

Thus, by using the compactness arguments, we can pass to the limits as η → 0 in (4.14), yields ρt + div(ρu) = 0, holds a.e. on (0, T ) × T3 , (ρu)t + div(ρu ⊗ u) + ∇P (ρ) − div(ρDu) + r0 u + r1 ρ|u|2 u − δρ∇3 ρ √  ρ − κρ∇( √ ) = ρ∇Φ, holds in the sense of distribution on (0, T ) × T3 , ρ

(5.4)

Φ = −4πG(ρ − 1), holds a.e. on (0, T ) × T3 . Similarly, due to the weak lower semi-continuity of convex functions, we can obtain the energy inequality and B-D entropy by passing to the limits in (4.15) and (4.16) as η → 0, we have ˆ ˆ Tˆ 1 2 ργ δ ρ|Du|2 + r0 |u|2 + r1 ρ|u|4 dxdt ≤ C(T ), ρu + + |∇ρ|2 dx + 2 aγ(γ − 1) 2 0 (5.5)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 19 and 1 2

ˆ

ˆ Tˆ ˆ ˆ γ 1 T 2 ∇ρ 2 2 ρ|∇u| + 2 |∇ρ 2 |2 ρ(u+ ) − r0 log ρdx + ρ 2 0 aγ 0 ˆ Tˆ ˆ Tˆ κ |2 ρ|2 + ρ|∇2 log ρ|2 ≤ C(T ), +δ 2 0 0

(5.6)

Passing to the limits as δ → 0.

6

Let (ρδ , uδ , Φ(ρδ )) denote the weak solutions at this level. Due to the energy inequality (5.5) and the B-D entropy (5.6), we have the following regularities: √

√ ρδ uδ ∈ L∞ (0, T ; L2 ), ρδ Duδ ∈ L2 (0, T ; L2 ), √ √ ∇ ρδ ∈ L∞ (0, T ; L2 ), δρδ ∈ L∞ (0, T ; H 3 ) ∩ L2 (0, T ; H 4 ), γ √ ργδ ∈ L∞ (0, T ; L1 ), ∇ρδ2 ∈ L2 (0, T ; L2 ), ρδ ∇uδ ∈ L2 (0, T ; L2 ), 1 1 1√ √ r0 uδ ∈ L2 (0, T ; L2 ), r14 ρδ4 uδ ∈ L4 (0, T ; L4 ), κ 2 ρδ ∇2 log ρδ ∈ L2 (0, T ; L2 ).

(6.1)

Next, we will proceed the compactness arguments in several steps.

6.1

Step 1: Convergence of

√ ρρδ .

In combination with the conservation of mass ρδ (t)L1 = ρδ (0)L1 and estimate in √ (6.1) gives ρδ ∈ L∞ (0, T ; H 1 ). Next, we notice that 1√ 1√ √ √ √ ρδ divuδ − uδ ∇ ρδ = ρδ divuδ − div(uδ ρδ ), ∂t ρδ = − 2 2 √ which yields ∂t ρδ ∈ L2 (0, T ; H −1 ), thanks to the Aubin-Lions Lemma, we have √ √ ρδ → ρ, strongly in L2 (0, T ; L2 ),

(6.2)

(6.3)

√ √ √ and hence yields ρδ → ρ a.e.. On the other hand, since ∇ ρδ is bounded in L∞ (0, T ; L2 ), by using the Sobolev embedding theorem, we have √ √  ρδ L∞ (L6 ) ≤ C∇ ρδ L∞ (L2 ) < +∞, so ρδ u δ =

√

3 √ ρδ,r0 ρδ uδ ∈ L∞ (0, T ; L 2 ),

(6.4)

3

3

which yields that div(ρδ uδ ) ∈ L∞ (0, T ; W −1, 2 ), immediately we have ∂t ρδ ∈ L∞ (0, T ; W −1, 2 ). 3 3 √ √ Furthermore, for ∇ρδ = 2 ρδ ∇ ρδ ∈ L∞ (0, T ; L 2 ), so we have ρδ is bounded in L∞ (0, T ; W 1, 2 ). 3 Then together with ∂t ρδ ∈ L∞ (0, T ; W −1, 2 ), thanks to the Aubin-Lions Lemma gives ρδ → ρ, strongly in C([0, T ]; Lp ), f or any p ∈ [1, 3),

(6.5)

and hence, we have ρδ → ρ a.e.. Similar to the proof of (3.25), The pressure P (ρδ ) ∈ 5 5 L 3 (0, T ; L 3 ), and up to subsequence, we have P (ρδ ) → P (ρ) a.e., and P (ρδ ) → P (ρ) strongly in L1 (0, T ; L1 ).

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 20

6.2

Step 3: Convergence of the momentum ρδ uδ .

Lemma 6.1. Up to a subsequence, the momentum mδ = ρδ uδ converges strongly in L2 (0, T ; Lq ) to some m(x, t) for all q ∈ [1, 32 ). In particular ρδ uδ → m a.e. f or (x, t) ∈ T3 × (0, T ). Note that we can define u(x, t) = m(x, t)/ρ(x, t) outside the vacuum set {x|ρ(x, t) = 0}. Proof. Since ∇(ρδ uδ ) = ∇ρδ uδ + ρδ ∇uδ √ √ √ √ = 2∇ ρδ ρδ uδ + ρδ ρδ ∇uδ ∈ L2 (0, T ; L1 ),

(6.6)

together with (6.4), yields ρδ uδ ∈ L2 (0, T ; W 1,1 ). In order to apply the Aubin-Lions Lemma, we also need to show ∂t (ρδ uδ ) is bounded in L2 (0, T ; H −s ), f or some constant s > 0, actually, from the momentum equation (5.4)2 , it’s easy to check that ∂t (ρδ uδ ) is bounded in L2 (0, T ; H −3 ). Hence, using the Aubin-Lions Lemma, the Lemma 6.1 is proved.

6.3

Step 4: Convergence of

√

ρδ u δ .

Recall the Lemma 6.1, we define velocity u(x, t) by setting u(x, t) = m(x, t)/ρ(x, t) when ρ(x, t) = 0 and u(x, t) = 0 when ρ(x, t) = 0, we have m(x, t) = ρ(x, t)u(x, t). Moreover, Fatou’s lemma yields ˆ Tˆ ˆ Tˆ ˆ Tˆ 4 4 ρu dxdt ≤ lim inf ρδ uδ dxdt ≤ lim inf ρδ u4δ dxdt, 0

0

δ→0

δ→0

0

1 4

hence, ρ u ∈ L4 (0, T ; L4 ). Since mδ → m a.e. and ρδ → ρ a.e., it’s easy to show that √ √ ρδ uδ → mδ / ρδ , a.e. in {ρ(x, t) = 0}, √ √ and for almost every (x, t) in {ρ(x, t) = 0}, we have ρδ uδ l|uδ |≤M ≤ M ρδ → 0, as a mat√ √ ter of fact, ρδ uδ l|uδ |≤M converges to ρul|u|≤M almost everywhere for (x, t). Meanwhile, √ ρδ uδ l|uδ |≤M is bounded in L∞ (0, T ; L6 ), using the Egorov’s theorem gives √ √ ρδ uδ l|uδ |≤M → ρul|u|≤M strongly in L2 (0, T ; L2 ). (6.7) Since √ √ √ √  ρδ uδ − ρu|2L2 (L2 ) ≤ 2 ρδ uδ l|uδ |≤M − ρul|u|≤M 2L2 (L2 ) √ √ + 2 ρδ uδ l|uδ |≥M |2L2 (L2 ) + 2 ρul|u|≥M |2L2 (L2 )

2 √ √ √ √ ≤ 2 ρδ uδ l|uδ |≤M − ρul|u|≤M |2L2 (L2 ) + 2 ( ρδ u2δ 2L2 (L2 ) +  ρu2 2L2 (L2 ) ) → 0, M

as δ → 0 and M → +∞. Thus we proved that √ √ ρδ uδ → ρu strongly in L2 (0, T ; L2 ).

(6.8)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 21

Step 5: Convergence of the terms ρδ Duδ , and ρδ ∇3 ρδ .

6.4

To deal with the diffusion term ρδ Duδ , recall (3.30), we have ˆ ˆ ˆ ˆ ˆ Tˆ 1 T 1 T div(ρδ Duδ )ϕ = (ρδ ujδ )∂ii ϕ + ∂i ρδ ujδ ∂i ϕ 2 2 0 0 0 ˆ ˆ ˆ Tˆ 1 1 T (ρδ uiδ )∂ij ϕ + ∂j ρδ uiδ ∂i ϕ, + 2 0 2 0 (6.9) ˆ Tˆ ˆ ˆ 1 T √ √ √ √ j j ( ρδ ρδ uδ )∂ii ϕ + ∂i ρδ ρδ uδ ∂i ϕ = 2 0 0 ˆ Tˆ ˆ ˆ 1 T √ √ √ √ i ( ρδ ρδ uδ )∂ij ϕ + ∂j ρδ ρδ uiδ ∂i ϕ, + 2 0 0 ´T ´ ´T ´ by using(6.3) and (6.8), we can show 0 div(ρδ Duδ )ϕ → 0 div(ρDu)ϕdxdt. Finally, we show the convergence of the high order term ρδ ∇3 ρδ : 2 5 √ 5 5 7 7 14 ρδ ∈ Since δ 14 ρδ  14 ≤ ρ  δρ  δ δ ∞ (L3 )  2 (H 4 ) ≤ C < +∞, which implies δ 3 L L 5 (H )

L

14

L 5 (0, T ; H 3 ). For any test function ϕ ∈ C ∞ ([0, T ]; T3 ), we have ˆ Tˆ ˆ Tˆ 3 ρδ ∇ ρδ ϕdxdt = −δ div(ρδ ϕ)2 ρδ dxdt, δ 0

0

we focus on the most difficult term ˆ Tˆ 1 √ 5 (∇ρδ )2 ρδ ϕdxdt| ≤ Cδ 7  δ2 ρδ L2 (L2 ) δ 14 ∇3 ρδ  |δ

L

0

14 5 (L2 )

ϕL7 (L∞ ) → 0,

as δ → 0. ´T ´ Similarly, we can handle the other terms from δ 0 div(ρδ ϕ)2 ρδ dxdt. Thus, we have ˆ Tˆ ρδ ∇3 ρδ ϕdxdt → 0, as δ → 0. δ 0

With all above compactness results, we can pass to the limits in (5.4) as δ → 0, we have ρt + div(ρu) = 0, holds in the sense of distribution on (0, T ) × T3 , √  ρ 2 (ρu)t + div(ρu ⊗ u) + ∇P (ρ) − div(ρDu) + +r0 u + r1 ρ|u| u − κρ∇( √ ) ρ

(6.10)

= ρ∇Φ, holds in the sense of distribution on (0, T ) × T , 3

Φ = −4πG(ρ − 1), holds a.e. on (0, T ) × T3 . Furthermore, due to the weak lower semi-continuity of the convex functions ,we can obtain the following energy inequality and B-D entropy by passing to the limits as δ → 0: ˆ Tˆ ˆ 1 1 2 γ (6.11) ρu + ρ dx + (ρ|Du|2 + r0 |u|2 + r1 ρ|u|4 )dxdt ≤ C(T ), 2 aγ(γ − 1) 0 and ˆ ˆ Tˆ γ ∇ρ 2 1 2 κ 1 ρ(u+ ( ρ|∇u|2 + 2 |∇ρ 2 |2 + ρ|∇2 log ρ|2 )dxdt ≤ C(T ). ) − r0 log ρdx + 2 ρ 2 aγ 2 0 (6.12)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 22

7

Mellet-Vasseur type inequality for the weak solutions

In this section, we are going to derive the Mellet-Vasseur type inequality by following the idea in [31]. This inequality will help us obtain the strong convergence of √ ρu in L2 (0, T ; L2 ), therefor the damping term r1 ρ|u|2 u can be removed at end of paper. Note that the proof in this section is almost exactly the same as that in [31], for the completeness of the paper and the convenience of readers, we will give a sketch of proof in the appendix, for more details, the readers can refer to [31]. Lemma 7.1. Let (ρ, u, Φ) be weak solutions to (6.10), then if 43 < γ < 3, we have the following Mellet-Vasseur type inequality: ˆ ˆ 2 2 ρ(1 + |u| ) ln(1 + |u| )dx ≤ ψ(0) ρ0 (1 + |u0 |2 ) ln(1 + |u0 |2 )dx + C ˆ (7.1) √ + C(ψ(t)L∞ ) (ρ0 |u0 |2 + ργ0 + |∇ ρ0 |2 − r0 log− ρ0 )dx.

Passing to the limits as r0 → 0, r1 → 0

8

In this section, we aim at removing the damping term r0 u and r1 |u|2 u by passing to the limits as r0 = r1 → 0. Thanks to the energy estimates (6.11), B-D entropy (6.12) and the Mellet-Vasseur type inequality (7.1), we have the enough regularity to prove the stability of the weak solutions to the Navier-Stokes-Poisson equations, the process is similar as in section 6, so we omit the details here. Hence, we complete the proof of the main theorem 1.1.

A

Appendix: Sketch proof of Lemma 7.1

Then we show the main idea of the proof for Mellet-Vassure inequality in following several steps:

A.1

Renormalize the momentum equations

We renormalize the velocity by introducing the following two cut-off functions φm and φK : 1 1 , φm (ρ) = 0 f or any ρ < , m 2m (A.1) φK (ρ) ≥ 0 ∈ C ∞ (R), φK (ρ) = 1 f or any ρ < K, φK (ρ) = 0 f or any ρ > 2K. φm (ρ) ≥ 0 ∈ C ∞ (R), φm (ρ) = 1 f or any ρ >

We define v = φ(ρ)u = φm (ρ)φk (ρ)u, due to the energy estimates (6.11), B-D entropy (6.12) and Lemma 3.2, it is easy to check that the renomalized velocity v is bounded in L2 (0, T ; H 1 ). Then we introduce the following approximate function ϕn (u): ϕn (u) ∈ C 1 (T3 ), ϕn (u) ≥ 0, ϕn (u) = ϕ˜n (|u|2 ),

(A.2)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 23 where ϕ˜n (|u|2 ) is given on R+ by: 

ϕ˜n (y) =

⎧ ⎨ ⎩

1 1+y 1 − 1+y

0

if 0 < y ≤ n, if n ≤ y < Cn , if y ≥ Cn ,

(A.3)



with ϕ˜n (0) = 1, ϕ˜n (0) = 0, and Cn = e(1 + n)2 − 1. Here we gather the properties of the function ϕ˜n in the following Lemma: Lemma A.1. Let ϕn and ϕ˜n be defined as above. Then they satisfy: 

• a. |ϕ˜n (y)| ≤

1 1+y

for any n > 0 and any y ≥ 0.

• b. For any u ∈ T3 , we have 





ϕn (u) = 2(2ϕ˜n (|u|2 )u ⊗ u + Iϕ˜n (|u|2 )),

(A.4)

where I is 3 × 3 unit matrix. Moreover, for any given n > 0 and any u ∈ T3 , we have  (A.5) |ϕn (u)| ≤ 6 + 2 ln(1 + n). • c.

⎧ if 1 ≤ y ≤ n, ⎨ 1 + ln(1 + y)  1 + 2 ln(1 + n) − ln(1 + y) if n < y ≤ Cn , ϕ˜n (y) = ⎩ 0 if y ≥ Cn ,

(A.6)



and |ϕ˜n (y)| ≤ 1 + ln(1 + y). • d.

⎧ if 0 ≤ y < n, ⎨ (1 + y) ln(1 + y) 2(1 + ln(1 + n))y − (1 + y) ln(1 + y) + 2(ln(1 + n) − n) if n ≤ y < Cn , ϕ˜n (y) = ⎩ if y ≥ Cn , e(1 + n)2 − 2n − 2 (A.7) ϕ˜n (y) is a nondecreasing function with respect to y for any fixed n, and it is also a nondecreasing function with respect to n for any fixed y. ϕ˜n (y) → (1 + y) ln(1 + y) a.e. as n → ∞.

(A.8)

Proof. The proof is exactly the same as that in [30], here we omit the details. Multiplying φ(ρ) on both sides of the momentum equation, we have 

(ρv)t − ρuφ (ρ)ρt + div(ρu ⊗ v) − ρu ⊗ u∇φ(ρ) + φ(ρ)∇P (ρ)

√  ρ − div(φ(ρ)ρDu) + ρ∇φ(ρ)Du + r0 uφ(ρ) + r1 ρ|u|2 uφ(ρ) − κdiv(ρφ(ρ) √ I) ρ √ √ √ √ + κ ρ∇φ(ρ) ρ + 2κφ(ρ)∇ ρ ρ = ρφ(ρ)∇Φ(ρ).

If we define

(A.9)

√  ρ S = ρφ(ρ)(Du + κ √ I), and ρ 

F = ρ2 uφ (ρ)divu + ρ∇φ(ρ)Du + r0 uφ(ρ) + r1 ρ|u|2 uφ(ρ) √ √ √ + κ ρ∇φ(ρ) ρ + 2κφ(ρ)∇φ(ρ) ρ,

(A.10)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 24 



where we used −ρuφ (ρ)ρt − ρu ⊗ u∇φ(ρ) = ρ2 uφ (ρ)divu, then the above equation (A.9) can be rewritten as (ρv)t + div(ρu ⊗ v) − divS + F = ρφ(ρ)∇Φ − ρ∇P(ρ),

(A.11)

thanks to the energy estimates (6.11) and B-D entropy (6.12), we can verify: SL2 (0,T ;L2 ) ≤ C(K, m, κ) and F 

4

4

L 3 (0,T ;L 3 )

≤ C(K, m, κ).

Follow the idea in [31], we first define a test function ψ(t) ∈ C0∞ which is supported in (0, T ), and later we will extend this function to ψ(t) ∈ (−1, T ). We define a test function ϕ = ψ(t)ϕn (v), where f (t, x) = f ∗ ην (t, x), ην (t, x) is modifier on (0, T ) × T3 . Then we use it to test (A.11) to have ˆ

T

ˆ



ψ(t)ϕn (v)[(ρv)t + div(ρu ⊗ v) − divS + F ]dxdt ˆ Tˆ  ψ(t)ϕn (v)[ρφ(ρ)∇Φ − φ(ρ)∇P (ρ)]dxdt. =

0

(A.12)

0

Note that the terms on the left hand side of the equation (A.12) are the same as those in [31], so in this section when we derive the Mellet-Vasseur type inequality for the weak solutions, we only focus on the different terms: the pressure ∇P (ρ) and the poisson term ρ∇Φ.

A.2

Passing to the limits as ν → 0

For the left hand side of the equation (A.12), borrowing the results from [31], we have ˆ LHS → − 0

T

ˆ

ˆ



ψ(t) ρϕn (v) +

0

T

ˆ



ψ(t)ϕn (v)F +

ˆ

T

ˆ

0



ψ(t)S : ∇(ϕn (v)),

(A.13)

where we used the properties of modifier, Lebesgue’s Dominated Convergence theorem and the properties of the cut-off function ϕn (u), for more details, the readers can refer to [31]. Then we consider the terms on the right hand side, since 

|φ∇P (ρ)| = |φP ∇ρ| ≤

1 2 γ− 1 √ √ φρ 2 |∇ ρ| + 2bφρ 2 |∇ ρ| ∈ L∞ (0, T ; L2 ), a 1

1

where we used the B-D entropy and φργ− 2 ≤ C(K, m), φρ 2 ≤ C(K, m). so, we have φ∇P (ρ) → φ∇P (ρ) strongly in L2 (0, T ; L2 ) as ν → 0. Note that v → v a.e. and ϕn (v) ∈ C 1 (T3 ), we have 





ϕn (v) → ϕn (v) a.e. and ϕn (v) ∈ L∞ ((0, T ) × T3 ), so by the Lebesgue’s Dominated Convergence theorem, we have ˆ Tˆ ˆ Tˆ   ψ(t)ϕn (v)φ∇P (ρ)dxdt → − ψ(t)ϕn (v)φ∇P (ρ)dxdt − 0

0

(A.14)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 25 Next due to the classical elliptic theory, we have ∇ΦLp ≤ Cρ

3p

L 3+p

√ , since ∇ ρ ∈

L∞ (0, T ; L2 ) ⇒ ρ ∈ L∞ (0, T ; L3 ), then in accordance with the conversation of mass ρ ∈ L∞ (0, T ; L1 ), we have ∇ΦL∞ (0,T ;Lp ) ≤ C, 32 ≤ p < ∞. Similarly, by the Lesbegue’s Dominated Convergence Theorem, we have ˆ

T

ˆ

ˆ



ψ(t)ϕn (v)ρφ∇Φ →

0

T

ˆ



(A.15)

ψ(t)ϕn (v)ρφ∇Φ.

0

when the index ν → 0, in combination with (A.13), (A.14) and (A.15), we have ˆ

ˆ

T

−

ˆ



T

ˆ

ˆ



T

ˆ



ψ(t) ρϕn (v) + ψ(t)ϕn (v)F + ψ(t)S : ∇(ϕn (v)) 0 0 ˆ Tˆ  ψ(t)ϕn (v)(ρφ∇Φ − φ∇P (ρ)), =

0

(A.16)

0

where S and F are defined in (A.10) and test function ψ(t) ∈ D(0, T ). Now we need to consider the test function ψ(t) ∈ D(−1, T ). This process is the same as that in [31], the readers can refer to [31]. So we borrow the results directly here, for any test function ψ(t) ∈ D(−1, T ), we have ˆ

T

−

ˆ

ˆ Tˆ ˆ Tˆ    ψ(t) ρϕn (v) + ψ(t)ϕn (v)F + ψ(t)S : ∇(ϕn (v)) 0 0 ˆ Tˆ ˆ  ψ(t)ϕn (v)(ρφ∇Φ − φ∇P (ρ)), = ρ0 ψ(0)ϕn (v0 )dx +

0

(A.17)

0

Passing to the limits as m → ∞

A.3

In this section, we need to pass to the limits as m → ∞ with K, κ, n, r0 , r1 being fixed. We denote vm = φm φK u at this level. Due to the definition of the cut-off function φm (ρ), we have φm (ρ) → 1 a.e. for (t, x),   φm (ρ) → 0 a.e. for (t, x) and |ρφm | ≤ C for all ρ, and moreover, we also have vm → φK u strongly in L2 (0, T ; H 1 ), and vm → φK u a.e. for (t, x), since vm ∈ L2 (0, T ; H 1 ). As the left hand side of the equation (A.17) is exactly the same as it in [30], we omit the process and borrow the results directly, we have ˆ

T

LHS → −

ˆ



ψ (t)ρϕn (φK u) +

0

ˆ 0

T

ˆ

ˆ

and

ˆ



ψ(t)ϕn (φK u)F +

T

0

ˆ



ψ(t)S : ∇(ϕn (φK u)) (A.18)

ˆ ψ(0)ρ0 ϕn (φK (vm (0))dx →

ψ(0)ρ0 ϕn (φK (ρ0 )u0 )dx. 



Due to vm → φK u a.e. for (t, x), we have ϕn (vm ) → ϕn (φK u) a.e. for (t, x) and |ϕn (vm )| ≤ C(n), then in combination with ∇P (ρ) ∈ L∞ (0, T ; L2 ) and Lebesgue’s Dominated Convergence Theorem, we have 

ˆ 0

T

ˆ



ψ(t)ϕn (vm )(ρφm φK ∇Φ − φm φK ∇P (ρ)) →

ˆ

T 0

ˆ



ψ(t)ϕn (φK u)(ρφK ∇Φ − φK ∇P (ρ)) (A.19)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 26 Thus together with (A.18) and (A.19), we have the follows ˆ Tˆ ˆ Tˆ ˆ Tˆ    ψ (t)ρϕn (φK u) + ψ(t)ϕn (φK u)F + ψ(t)S : ∇(ϕn (φK u)) − 0 0 0 (A.20) ˆ Tˆ ˆ  ψ(t)ϕn (φK u)(ρφK ∇Φ − φK ∇P (ρ)), = ψ(0)ρ0 ϕn (φK (ρ0 )u0 )dx + 0

√  ρ S = φK ρ(Du + κ √ I), ρ

where

√  √  F = ρ2 uφK divu + r0 φK u + r1 φK |u|2 u + κ ρφK ∇ρ ρ √ √ √ √  + 2κφK ∇ ρ ρ + 2 ρDuρφK ∇ ρ.

A.4

(A.21)

Passing to the limits as κ → 0 and K → ∞

In this section, we need to pass to the limits as κ → 0 and K → ∞ with n, r0 , r1 3 being fixed. Follow the [30], we assume that K = κ− 4 , thus K → ∞ when κ → 0. Due to the definition of φK , we have 

φK → 1, φK → 0,



as K → ∞, and |φK | ≤

2 , f or K ≤ ρ ≤ 2K, K 3

which yields vκ = φK u → u, a.e. f or (t, x), as K = κ− 4 → ∞, κ → 0. Then we consider the pressure term: ˆ Tˆ  ψ(t)ϕn (vκ ) · ∇P (ρκ )φK (ρκ ) 0 ˆ Tˆ ˆ Tˆ   ψ(t)P (ρκ )φK (ρκ )ϕn : ∇vκ − ψ(t)P (ρκ )ϕn (vκ ) · ∇φK (ρκ ) =− 0

(A.22)

0

= P1 + P2 , first, we control P2 as follows ˆ Tˆ 1  (| ργκ + bρκ )|ϕn ||∇φK (ρκ )| |P2 | ≤ ψ(t)L∞ aγ 0 1 √  γ+ 1 ≤ C(n, ψL∞ )κ− 4  ρκ φK L∞ (0,T ;L∞ ) ρκ 4  + C(n, ψ ≤

L∞

3 2

√

4 4 L 3 (0,T ;L 3 )

1

1

κ 4 ∇ρκ4 L4 (0,T ;L4 )

(A.23)



)ρκ L∞ (0,T ;L2 ) ∇ ρκ L∞ (0,T ;L2 ) φK L∞ (0,T ;L∞ )

1 3 C(n, ψL∞ ) − 1 C(n, ψL∞ ) √ √ κ 4+ ∇ ρκ 2L∞ (0,T ;L2 ) ≤ C(κ 8 + κ 4 ) → 0 K K

as κ → 0, where we need the fact 43 (γ + 14 ) ≤ 5γ 3 for any γ > 1. For P1 , we have ˆ Tˆ  ψ(t)P (ρκ )φK (ρκ )ϕn : ∇(φK uκ ) P1 = − 0 ˆ Tˆ ˆ Tˆ    2 ψ(t)P (ρκ )φK (ρκ )ϕn : ∇uκ − ψ(t)P (ρκ )φK (ρκ )ϕn : (φK ∇ρκ ⊗ uκ ) =− 0

0

= P11 + P12 , (A.24)

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 27 and P12 can be controlled similarly as p2 , we have ˆ Tˆ 1 3  (ργκ + ρκ )|φK ||∇ρκ | ≤ C(κ 8 + κ 4 ) → 0. |P12 | ≤ C(n, ψL∞ )

(A.25)

0

Since







ϕn (vκ ) : ∇uκ = 4∇uκ ϕ˜n (|vκ |2 )vκ ⊗ vκ + 2divuκ ϕ˜n (|vκ |2 ), we have ˆ Tˆ   ψ(t)P (ρκ )φ2K [4∇uκ ϕ˜n (|vκ |2 )vκ ⊗ vκ + 2divuκ ϕ˜n (|vκ |2 )]dxdt| |P11 | = | 0 ˆ Tˆ 1  ψ(t)( ργκ + bρκ )φ2K (4|∇uκ | + 2|ϕ˜n (vκ2 )||divuκ |) ≤ aγ 0 ˆ Tˆ ˆ Tˆ   2 + ρκ ) ≤ C(ψL∞ ) ρκ |∇vκ | + (ρ2γ−1 κ 0 0 ˆ Tˆ ˆ Tˆ   2 2 +ε ψ(t)ϕ˜n (|vκ | )ρκ |Duκ | + C(ε, ψL∞ ) ϕ˜n (|vκ |2 )(ρ2γ−1 + ρκ ) κ 0 0 ˆ Tˆ ˆ Tˆ   ψ(t)ϕ˜n (|vκ |2 )ρκ |Duκ |2 + C(ε, ψL∞ ) (1 + ϕ˜n (|vκ |2 ))(ρ2γ−1 + ρκ ) + E0 , ≤ε κ 0

0

(A.26) + |∇ ρ0 − r0 log− ρ0 )dx. The first where ε << 1 and E0 = C(ψL∞ ) (ρ0 |u0 + right hand side term of (A.26) can be absorbed by the dispersion term (see [31]), and due to the Lebesgue’s Dominated Convergence Theorem, we have ˆ Tˆ ˆ Tˆ   2 2γ−1 (A.27) + ρκ ) → (1 + ϕ˜n (|vκ | ))(ρκ (1 + ϕ˜n (|u|2 ))(ρ2γ−1 + ρ), ´

ργ0

|2

0

√

|2

0

as κ → 0. Similarly, we also have ˆ Tˆ ˆ Tˆ   ψ(t)ϕn (φK u)ρκ φK ∇Φ(ρκ ) → − ψ(t)ϕn (u)ρ∇Φ(ρ), − 0 ˆ 0 ˆ ψ(0)ρ0 ϕn (φK (ρ0 )u0 )dx → ψ(0)ρ0 ϕn (u0 )dx.

(A.28)

Then borrowing the results from [31] for terms on the left hand side of (A.20) and together with (A.23)-(A.28), we have ˆ Tˆ ˆ √  − ψ (t)ρϕn (u) ≤ C(ψ(t)L∞ ) (ρ0 |u0 |2 + ργ0 + |∇ ρ0 |2 − r0 log− ρ0 )dx 0 ˆ Tˆ ˆ  (A.29) (1 + ϕ˜n (|u|2 ))(ρ2γ−1 + ρ) + ψ(0) ρ0 ϕn (u0 )dx + C(ψ(t)L∞ ) ˆ + 0

A.5

T

ˆ

0



ψ(t)ϕn (u)ρ · ∇Φ.

Passing to the limits as n → ∞

The objective of this section is to derive the Mellet-Vasseur inequality for the weak solutions by passing to the limits as n → ∞. First, thanks to the Monotone Convergence

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 28 Theorem , we have ˆ Tˆ ˆ  ψ (t)ρϕn (u) → − − 0

T

ˆ



ψ (t)ρ(1 + |u|2 ) ln(1 + |u|2 ),

(A.30)

0

as n → ∞. Due to part c of Lemma A.1, the third term on the right hand side of the inequality (A.29) can be controlled as ˆ Tˆ ˆ Tˆ  2 2γ−1 (1 + ϕ˜n (|u| ))(ρ (2 + ln(1 + |u|2 ))(ρ2γ−1 + ρ) + ρ) ≤ C 0 0 ˆ ˆ T ˆ 2 δ  1− δ   2 δ 2 δ 2 2 dt ρ 2 + ln(1 + |u| ) dx (ρ2γ−1− 2 ) 2−δ dx ≤C (A.31) 0 ˆ Tˆ ρ(1 + |u|2 )dxdt +C 0 √ ≤ C(ρL∞ (0,T ;L1 ) +  ρuL∞ (0,T ;L2 ) + ργ  53 5 ) ≤ C 3 L (0,T ;L )

6−2γ 2 ≤ 5γ where we require (2γ − 1 − 2δ ) 2−δ 3 ⇒ 0 < δ ≤ 5γ−3 , for any δ ∈ (0, 2), consequently, we need 43 < γ < 3. For the poisson term, we have ˆ Tˆ ˆ Tˆ   2 ψ(t)ϕn (|u| )ρ · ∇Φ ≤ ψ(t)ϕ˜n (|u|2 )|u|ρ|∇Φ| 0 0 ˆ Tˆ ˆ Tˆ 1 ψ(t)(1 + ln(1 + |u|2 ))|u|ρ|∇Φ| ≤ C(ψL∞ ) (1 + |u|2 ) 4 |u|ρ|∇Φ| ≤ 0 0 √ √ ≤ C(ψL∞ )[ ρuL∞ (0,T ;L2 )  ρL∞ (0,T ;L6 ) ∇ΦL∞ (0,T ;L3 ) (A.32) 3

1

+ (ρu2 ) 4  ∞ 4 ρ 4 L∞ (0,T ;L12 ) ∇ΦL∞ (0,T ;L6 ) ] L (0,T ;L 3 ) √ √ ≤ C(ψL∞ )[ ρuL∞ (0,T ;L2 ) ∇ ρL∞ (0,T ;L2 ) ρ ∞ 3 2 √

√

3 2

L (0,T ;L )

1 2

+  ρuL∞ (0,T ;L2 ) ∇ ρL∞ (0,T ;L2 ) ρL∞ (0,T ;L2 ) ] ≤ C With (A.30)-(A.32), letting n → ∞, we have ˆ Tˆ ˆ  ψ (t)ρ(1 + |u|2 ) ln(1 + |u|2 ) ≤ ψ(0) ρ0 (1 + |u0 |2 ) ln(1 + |u0 |2 )dx + C − 0 ˆ √ + C(ψ(t)L∞ ) (ρ0 |u0 |2 + ργ0 + |∇ ρ0 |2 − r0 log− ρ0 )dx. Taking

⎧ ⎨ 1 ψ(t) =

⎩

1 2

0

−

if t < t˜ − 2ε if t˜ − 2ε ≤ t ≤ t˜ + if t > t˜ + 2ε

t−t˜ ε

then for any t˜ ≥ 2ε , (A.33) gives 1 ε

ˆ

t˜+ 2ε

t˜− 2ε

(A.34)

ˆ

ˆ

ρ(1 + |u|2 ) ln(1 + |u|2 ) ≤ ψ(0) ˆ + C(ψ(t)L∞ )

ε 2

(A.33)

(ρ0 |u0 | + 2

ργ0

ρ0 (1 + |u0 |2 ) ln(1 + |u0 |2 )dx + C √

(A.35)

+ |∇ ρ0 | − r0 log− ρ0 )dx. 2

Y.Ye and C.Dou global weak solutions to Compressible Navier-Stokes-Poisson equations 29 Then the Mellet-Vasseur type inequality for the weak solutions can be obtained after using the Lebesgue point Theorem, we have ˆ ˆ 2 2 ρ(1 + |u| ) ln(1 + |u| )dx ≤ ψ(0) ρ0 (1 + |u0 |2 ) ln(1 + |u0 |2 )dx + C ˆ (A.36) √ + C(ψ(t)L∞ ) (ρ0 |u0 |2 + ργ0 + |∇ ρ0 |2 − r0 log− ρ0 )dx. Acknowledgements. This research was partially supported by NSFC (Grant Nos. 11401395,11471134) and BNSF (Grant No. 1154007).

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