Global well-posedness of classical solutions to a fluid–particle interaction model in R3

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ScienceDirect J. Differential Equations 263 (2017) 8666–8717 www.elsevier.com/locate/jde

Global well-posedness of classical solutions to a fluid–particle interaction model in R3 Shijin Ding a , Bingyuan Huang b,∗ , Huanyao Wen c,∗ a School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China b School of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, China c Department of Mathematics, South China University of Technology, Guangzhou 510641, China

Received 20 May 2016; revised 23 August 2017

Abstract We consider the Cauchy problem of a fluid–particle interaction model in R3 , namely, compressible Navier–Stokes equations coupled with Smoluchowski equation. When the initial data (ρ0 , u0 , η0 ) is of small energy around steady state (ρ , 0, η ), the global well-posedness and large-time behavior of classical solutions are investigated. Vacuum is allowed. © 2017 Elsevier Inc. All rights reserved. MSC: 35Q30; 76N10; 46E35 Keywords: Compressible fluid–particle interaction model; Global classical solution; Large-time behavior; Decay rates

1. Introduction The form of a fluid–particle interaction model called as Navier–Stokes–Smoluchowski equations in [2,5,6] is governed by the Smoluchowski equation coupled with the Navier–Stokes equations for a compressible fluid in R3 as follows: * Corresponding authors.

E-mail addresses: [email protected] (S. Ding), [email protected] (B. Huang), [email protected], [email protected] (H. Wen). http://dx.doi.org/10.1016/j.jde.2017.08.048 0022-0396/© 2017 Elsevier Inc. All rights reserved.

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

ρt + ∇ · (ρu) = 0, ρut + ρu · ∇u + ∇(pF + η) − μu − λ∇(∇ · u) = −(η + βρ)∇, ηt + ∇ · (η(u − ∇)) = η,

8667

(1.1) (1.2) (1.3)

where the density of the fluid ρ ≥ 0, the fluid velocity field u = (u1 , u2 , u3 ), and the density of the particles in the mixture η ≥ 0 is related to the probability distribution function f (t, x, ξ ) in the macroscopic description through the relation  η(t, x) =

f (t, x, ξ )dξ. R3

Here ∇ · (= div) denotes the spatial divergence operator on R3 , the function pF denotes the pressure of the fluid, given by pF = pF (ρ) = aρ γ ,

a > 0,

γ > 1.

(1.4)

And the time independent external potential  = (x) : R3 → R+ is the effects of gravity and buoyancy, β is a constant reflecting the differences in how the external force affects the fluid and the particles, λ and μ are constant viscosity coefficients satisfying the physical condition: μ > 0,

2 λ + μ ≥ 0. 3

(1.5)

Without the dynamic viscosity terms in (1.2), this system was derived by Carrillo and Goudon [5], in which they introduced the flowing regime and the bubbling regime with respect to two different scalings and investigated the stability and asymptotic limits. The system (1.1)–(1.3) is known as the bubbling regime. When the viscous effects are taken into account, there are some previously relevant works on the system (1.1)–(1.3). For the global existence of weakly dissipative solutions as well as their weak-strong uniqueness and low Mach number limits in high dimensions, please refer to Ballew–Trivisa’s work [3], Carrillo et al.’s work [6] and Ballew’s work [1], respectively. In particular, Carrillo et al. in their work [6] proved that the weak solutions exist globally in time and that the weak solutions converge to a stationary solution as time goes to ∞. When η is zero, the system (1.1)–(1.3) is reduced to compressible Navier–Stokes equations for isentropic flow. Even for the compressible Navier–Stokes equations for isentropic flow, the uniqueness of global weak solutions derived by P.L. Lions (refer to [17]. See also [9,14]) is still open. Thus, it is natural to study some regular solutions that can ensure the uniqueness. Refer to [2,11] for the local well-posedness of strong solutions in a bounded domain and classical solutions in R3 , respectively. In one dimension, the global well-posedness of classical solutions with large initial data and vacuum was derived by Fang et al. [8]. In three dimensions, Chen, the first author, and Wang [7] established the existence (also uniqueness) theory of global classical solutions under some smallness assumptions on both the external potential and the perturbation of the initial data in a neighborhood of a stationary profile (ρ∗, 0, 0) in some Sobolev spaces. The method in their proofs in [7] relies on the analysis of the corresponding linearized system and the time-decay estimates of η in L2 . Thus, that the initial density ρ0 has positive lower bound is quite essential, i.e., no vacuum at any point.

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In this paper, we continue to study the global well-posedness of regular solutions to (1.1)–(1.3) with vacuum as well as large-time behavior in R3 . The initial data is of small perturbation around a stationary profile. Here one of the cases for the stationary profile seems more physical, i.e., (ρ , 0, η ) where the stationary density of the particles η > 0 or η = 0. Several challenges are associated with the problem. • The momentum equations are degenerate at vacuum points. • The domain is unbounded. Here we have to use the time-decay estimate of η to get the basic energy estimate when η = 0 (see Lemma 4.2). • A stability problem of the new stationary profile (ρ , 0, η ) where η > 0 is considered. This case seems more physical, since the particles will not disappear generally. In this case, the time-decay estimate of η is not necessary to get the basic energy estimate (see Lemma 7.1). More precisely, we consider the initial value problem: (ρ, u, η)(x, 0) = (ρ0 , u0 , η0 ),

in

R3 ,

(1.6)

and the far field behavior (ρ, u, η)(x, t) → (ρ ∞ , 0, η∞ )

as |x| → ∞,

for some constant vector (ρ ∞ , 0, η∞ ) satisfying ρ ∞ > 0, η∞ ≥ 0. As for the stationary problem of (1.1)–(1.7), we consider two cases in the following. Case one: If η∞ = 0, referred to [7], the steady solution (ρ∗ , 0, 0) is determined by  ∇pF (ρ∗ ) = −ρ∗ β∇, ρ∗ (x) → ρ ∞ as |x| → ∞.

(1.7)

(1.8)

Similar to the steady solution in [15], the stationary problem (1.8) gives that ρ∗ ρ∞

pF (ρ) dρ + β(x) = 0, ρ

(1.9)

and 

∞ γ −1

ρ∗ (x) = (ρ )

 1 γ −1 γ −1 − >0 β aγ

(1.10)

satisfying sup  < x∈R3

β −1 aγ ∞ γ −1 , (ρ ) γ −1

β > 0.

Case two: if η∞ > 0, the steady solution (ρ , 0, η ) is determined by ⎧ ⎪ ⎨∇(pF (ρ ) + η ) = −(η + βρ )∇, −∇ · (η ∇) = η , ⎪ ⎩ ρ (x) → ρ ∞ , η (x) → η∞ as |x| → ∞,

(1.11)

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which has a especial steady solution satisfying ⎧ ⎪ ⎨∇pF (ρ ) = −βρ ∇, ∇η = −η ∇, ⎪ ⎩ ρ (x) → ρ ∞ , η (x) → η∞

(1.12) |x| → ∞.

as

Thus, the stationary problem (1.12) gives that ⎧

⎪ ⎪ ⎨ρ (x) = ρ∗ = (ρ ∞ )γ −1 −



γ −1 aγ β

1 γ −1

, (1.13)

(x) = η∞ e− ,

η ⎪ ⎪ ⎩ ρ (x) = ρ∗ → ρ ∞ , η (x) → η∞

as |x| → ∞.

Throughout this paper, we denote f˙  ft + u · ∇f.

(1.14)

The initial energy is defined as:  

 1 ρ0 |u0 |2 + G(ρ0 ) dx, 2

(1.15)

 1 1 2 ∞ − 2 ρ0 |u0 | + G(ρ0 ) + |η0 − η e | dx, 2 2

(1.16)

C0 = and C˜ 0 =

 

where G denotes the potential energy density given by ρ r G(ρ) 

pF (s) dsdr = ρ s

ρ∗ ρ∗

ρ

pF (s) − pF (ρ∗ ) ds. s2

ρ∗

For 1 ≤ r ≤ ∞, we make the following notations for the standard homogeneous and inhomogeneous Sobolev spaces (for more details also see in [10]). (1) Lr = Lr (R3 ), D k,r = {v ∈ L1loc (R3 ) : v D k,r ≡ ∇ k v Lr < ∞}.  (2) W k,r = Lr ∩ D k,r , H k = W k,2 , D k = D k,2 , D 1 = u ∈ L6 ∇u L2 < ∞ . Remark 1.1. (Proposition 1.1 in [15]) Let  ∈ H 3 satisfy (1.11), then there exists a unique solution ρ∗ = ρ∗ (x) for the stationary problem (1.8) satisfying (ρ∗ − ρ ∞ , pF (ρ∗ ) − pF (ρ ∞ )) ∈ H 3 ,

0 < ρ ≤ inf ρ∗ ≤ sup ρ∗ ≤ ρ¯ < ∞, x∈R3

(1.17)

x∈R3

∇ρ∗ H 2 ≤ C,

(1.18)

where ρ, ρ¯ and C are positive constants depending only on ρ ∞ , a, γ , β, inf , sup ,  H 3 . x∈R3

x∈R3

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Remark 1.2. From the direct calculation, it is clear that (ρ − ρ∗ )2 ≤ c1 (ρ, ˜ ρ)G(ρ) ¯

(1.19)

˜ ρ). ¯ for any ρ ∈ [0, ρ] ˜ and a positive constant c1 (ρ, 2. Main results Theorem 2.1. Assume that η∞ = 0,  ∈ H 4 satisfies (1.11), ρ∗ = ρ∗ (x) is the steady state in (1.8), and the initial data (ρ0 , u0 , η0 ) satisfy 

ρ0 |u0 |2 + G(ρ0 ) ∈ L1 ,

(u0 , η0 ) ∈ D 1 ∩ D 3 ,

0 ≤ inf ρ0 ≤ sup ρ0 ≤ ρ, ˜

∇u0 2L2 ≤ M1 ,

(ρ0 − ρ ∞ , pF (ρ0 ) − pF (ρ ∞ )) ∈ H 3 ,

∇η0 2L2 ≤ M2 ,

3

∇ 3L2 + η0 Lp ≤ ε 2 , (2.1)

for given numbers M1 , M2 > 0 (not necessarily small), ρ˜ ≥ ρ¯ + 1, C0 ≤ ε, ε ∈ (0, 1), p ∈ [1, 65 ), and that the compatibility condition holds −μ u0 − λ∇(∇ · u0 ) + ∇(pF (ρ0 ) + η0 ) + η0 ∇ = −ρ0 (β∇ + g)

(2.2)

1/2

for some g ∈ D 1 with ρ0 g ∈ L2 . Then there exists a positive constant ε¯ depending on ˜ inf ,  H 3 , M1 and M2 such that if μ, λ, ρ ∞ , a, γ , β, ρ, x∈R3

ε ≤ ε¯ ,

(2.3)

the Cauchy problem (1.1)–(1.7) has a unique global classical solution (ρ, u, η) in R3 × (0, ∞) satisfying for any 0 < τ < T < ∞, 0 ≤ ρ(x, t) ≤ 2ρ, ˜

x ∈ R3 , t ≥ 0,

⎧ ⎪ (ρ − ρ ∞ , pF − pF (ρ ∞ )) ∈ C([0, T ]; H 3 ), ⎪ ⎪ ⎪ ⎪ ⎨(u, η) ∈ C([0, T ]; H 3 ) ∩ L2 (0, T ; D 4 ) ∩ L∞ (τ, T ; D 4 ), ⎪ (ut , ηt ) ∈ L∞ (0, T ; H 1 ) ∩ L2 (0, T ; H 2 ) ∩ L∞ (τ, T ; D 2 ) ∩ H 1 (τ, T ; D 1 ), ⎪ ⎪ ⎪ ⎪ ⎩√ρu ∈ L∞ (0, T ; L2 ),

(2.4)

(2.5)

t

and the following large-time behavior:  lim

t→∞

(|ρ − ρ∗ |q + ρ 1/2 |u|4 )dx + ∇u(·, t) Lr¯ + ∇η(·, t) Lr¯ = 0,

holds for r¯ ∈ [2, 6), q ∈ (2, ∞).

(2.6)

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Remark 2.1. The solution obtained in Theorem 2.1 is classical. It should be emphasized that the pressure term containing ∇η in (1.2) make us difficult to obtain the standard basic energy estimate for solutions (ρ, u, η) to the system (1.1)–(1.8). Fortunately, the time-decay estimate T of η 2L2 in (4.9) help us get the time-independent bound of 0 η 2L2 dt, which will plays an important role in closing the a priori estimate in (4.17). Theorem 2.2. Under the assumptions of Theorem 2.1, there exists a positive constant C¯ depending on ε but not depending on the time t such that − 32 ( p1 − 12 )

¯ + t)

∇ k η L2 ≤ C(1

− 32 ( p1 − 12 )

¯ + t)

η Lq ≤ C(1

,

,

f or

k = 0, 1, q ∈ [2, 6].

Theorem 2.3. Assume that η∞ > 0,  ∈ H 4 satisfies (1.11), (η∞ − η∞ e− ) ∈ H 3 , (ρ , η ) = (ρ (x), η (x)) is the steady state in (1.12) and the initial data (ρ0 , u0 , η0 − η∞ e− ) satisfy ⎧ 2 ∞ − 2 1 ∞ − 1 3 ⎪ ⎨ρ0 |u0 | + G(ρ0 ) + |η0 − η e | ∈ L , (u0 , η0 − η e ) ∈ D ∩ D , ˜

∇u0 2L2 ≤ M˜ 1 , (ρ0 − ρ ∞ , pF (ρ0 ) − pF (ρ ∞ )) ∈ H 3 , 0 ≤ inf ρ0 ≤ sup ρ0 ≤ ρ, ⎪ ⎩ ∇(η − η∞ e− ) 2 ≤ M˜ , ∇

∞ − η∞ e− ≤ ν 12 , 3 + η 3 0

L2

2

(2.7)

L

L3 ∩L 2

for given numbers M˜ 1 , M˜ 2 > 0 (not necessarily small), ρ˜ ≥ ρ¯ + 1, C˜ 0 ≤ ν, ν ∈ (0, 1), and that the compatibility condition holds in (2.2) with η0 replaced by (η0 − η∞ e− ). Then there exists a positive constant ν˜ depending on μ, λ, ρ ∞ , η∞ , a, γ , β, ρ, ˜ inf ,  H 3 , η∞ − η∞ e− H 2 , x∈R3

M˜ 1 and M˜ 2 such that if ν ≤ ν˜ ,

(2.8)

the Cauchy problem (1.1)–(1.7) has a unique global classical solution (ρ, u, η) in R3 × (0, ∞) satisfying for any 0 < τ < T < ∞, 0 ≤ ρ(x, t) ≤ 2ρ, ˜

x ∈ R3 , t ≥ 0,

⎧ (ρ − ρ ∞ , pF − pF (ρ ∞ )) ∈ C([0, T ]; H 3 ), ⎪ ⎪ ⎪ ⎨(u, η − η∞ e− ) ∈ C([0, T ]; H 3 ) ∩ L2 (0, T ; D 4 ) ∩ L∞ (τ, T ; D 4 ), ⎪ (ut , ηt ) ∈ L∞ (0, T ; H 1 ) ∩ L2 (0, T ; H 2 ) ∩ L∞ (τ, T ; D 2 ) ∩ H 1 (τ, T ; D 1 ), ⎪ ⎪ ⎩√ ρut ∈ L∞ (0, T ; L2 ),

(2.9)

(2.10)

and the large-time behavior of (2.6) holds with η replaced by (η − η∞ e− ). The remaining part of this paper is organized as follows. In section 3, we state some useful lemmas which will be used in the next sections. In section 4, we derive the lower-order a priori estimates on classical solutions of the system (1.1)–(1.8). In section 5, the time-dependent estimates on higher-order estimates of the system (1.1)–(1.8) are established, which are needed for the global existence of classical solutions. In section 6, Theorem 2.1 and Theorem 2.2 will be proved. Finally, Theorem 2.3 is obtained for the system (1.1)–(1.7) and (1.12) in section 7.

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3. Preliminaries First, we start with the local existence and uniqueness of classical solutions in [11], which are needed to guarantee a priori assumption (4.6). Lemma 3.1 ([11]). Assume that (ρ0 − ρ ∞ , pF 0 − pF∞ ) ∈ H 3 , ρ ∞ ∈ R+ , ρ0 ≥ 0, η0 ≥ 0, u0 ∈ D 1 ∩ D 3 , η0 ∈ H01 ∩ D 3 ,  ∈ H 4 , (3.1) where the pressure pF 0 = aρ0 , pF∞ = a(ρ ∞ )γ , and the compatibility condition satisfies (2.2). Then there exist a small time T∗ > 0 and a unique classical solution (ρ, u, η) to the Cauchy problem (1.1)–(1.7) with η∞ = 0 on R3 × (0, T∗ ] such that γ

⎧ ⎪ (ρ − ρ ∞ , pF − pF (ρ ∞ )) ∈ C([0, T∗ ]; H 3 ), u, ∈ C([0, T∗ ]; D01 ∩ D 3 ) ∩ L2 (0, T∗ ; D 4 ), ⎪ ⎪ ⎪ ⎪ ⎪ η ∈ C([0, T∗ ]; H01 ∩ D 3 ) ∩ L2 (0, T∗ ; D 4 ), ut ∈ L∞ (0, T∗ ; D01 ) ∩ L2 (0, T∗ ; D 2 ), ⎪ ⎪ ⎪ ⎪ √ √ ⎪ ⎪ηt ∈ L∞ (0, T∗ ; H01 ) ∩ L2 (0, T∗ ; D 2 ), ρut ∈ L∞ (0, T∗ ; L2 ), ρutt ∈ L2 (0, T∗ ; L2 ), ⎪ ⎪ ⎪ 1 1 ⎪ ⎨(t 1/2 u, t 1/2 η) ∈ L∞ (0, T∗ ; D 4 ), (t 2 ut , t 2 ηt ) ∈ L∞ (0, T∗ ; D 2 ), t 12 utt ∈ L2 (0, T∗ ; D 1 ), 0 1 2 (0, T ; H 1 ), t 12 √ρu ∈ L∞ (0, T ; L2 ), (tu , tη ) ∈ L∞ (0, T ; D 3 ), ⎪ 2 η ∈ L t ⎪ tt ∗ tt ∗ t t ∗ 0 ⎪ ⎪ ⎪ ∞ (0, T ; D 1 ) ∩ L2 (0, T ; D 2 ), tη ∈ L∞ (0, T ; H 1 ) ∩ L2 (0, T ; D 2 ), ⎪ ⎪ ∈ L tu tt ∗ ∗ tt ∗ ∗ ⎪ 0 0 ⎪ ⎪ 3 3 √ ⎪ ∞ 2 ∞ 2 ⎪ 2 2 t ρuttt ∈ L (0, T∗ ; L ), (t utt , t ηtt ) ∈ L (0, T∗ ; D ), ⎪ ⎪ ⎪ ⎪ 3 3 ⎩ 32 √ t ρuttt ∈ L∞ (0, T∗ ; L2 ), t 2 uttt ∈ L2 (0, T∗ ; D01 ), t 2 ηttt ∈ L2 (0, T∗ ; H01 ). (3.2) Lemma 3.2 (Gagliardo–Nirenberg). ([16]) For p ∈ [2, 6], q ∈ (1, ∞), and r ∈ (3, ∞), there exists some generic constant C > 0 which may depend on q, r such that for f ∈ H 1 (R3 ) and g ∈ Lq (R3 ) ∩ D 1,r (R3 ), we have p

(6−p)/2

f Lp ≤ C f L2

 C R3

g

q(r−3)/(3r+q(r−3))

≤ C g Lq

(3p−6)/2

∇f L2

(3.3)

,

3r/(3r+q(r−3))

∇g Lr

.

(3.4)

Next, the following Zlotnik inequality will be used to get the uniform (in time) upper bound of ρ. Lemma 3.3 ([18]). Let the function y satisfy y  (t) = g(y) + b (t) on [0, T ],

y(0) = y 0 ,

with g ∈ C(R) and y, b ∈ W 1,1 (0, T ). If g(∞) = −∞ and b(t2 ) − b(t1 ) ≤ N0 + N1 (t2 − t1 )

(3.5)

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for all 0 ≤ t1 < t2 ≤ T with some N0 ≥ 0 and N1 ≥ 0, then   y(t) ≤ max y 0 , ζ + N0 < ∞ on [0, T ],

(3.6)

where ζ is a constant such that g(ζ ) ≤ −N1

for

ζ ≥ ζ.

(3.7)

Finally, to estimate ∇u L∞ and ∇ρ L2 ∩L6 , we state the following Beal–Kato–Majda type inequality which was proved in [4,13]. Lemma 3.4. For 3 < q˜ < ∞, there is a constant C(q) ˜ such that the following estimate holds for all ∇u ∈ L2 (R3 ) ∩ D 1,q¯ (R3 ),  

∇u L∞ (R3 ) ≤ C ∇ · u L∞ (R3 ) + ω L∞ (R3 ) log(e + ∇ 2 u Lq˜ (R3 ) ) + C ∇u L2 (R3 ) + C. (3.8) 4. Time-independent lower-order estimates of (1.1)–(1.8) In this section, we will establish the time-independent (weighted) necessary estimates and the uniform upper bound of density, to extend the local classical solution guaranteed by Lemma 3.1. Thus, let T > 0 be a fixed time and (ρ, u, η) be the smooth solution to (1.1)–(1.7) on R3 × (0, T ] in the class (3.2) with smooth initial data (ρ0 , u0 , η0 ) satisfying (2.1)–(2.2). To estimate this solution, we define A1 (T )  sup η 2L2 ,

(4.1)

t∈[0,T ]

  A2 (T )  sup

t∈[0,T ]

 T  ρ|u|2 |∇u|2 dxdt, dx + G(ρ) + 2 0

 T 

 2 2 σ ρ|u| ˙ 2 + |ηt |2 + |∇ 2 η|2 dxdt, A3 (T )  sup σ ∇u L2 + ∇η L2 + t∈[0,T ]

t∈[0,T ]

(4.3)

0

 A4 (T )  sup

(4.2)

σ

3



ρ|u| ˙ + |ηt | + |∇ η| 2

2

2

2



T  dx +

 σ 3 |∇ u| ˙ 2 + |∇ηt |2 dxdt, (4.4)

0

and A5 (T )  sup ( ∇u 2L2 + ∇η 2L2 ), t∈[0,T ]

where σ (t)  min {1, t}. We can prove the following key a priori estimates on (ρ, u, η).

(4.5)

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Proposition 4.1. Under the conditions of Theorem 2.1, there exists another positive constant K ˜ β, inf ,  H 3 , M1 and M2 such that if (ρ, u, η) is a smooth depending on μ, λ, ρ ∞ , a, γ , ρ, x∈R3

solution of (1.1)–(1.7) on R3 × (0, T ] satisfying ⎧ ⎨0 ≤ ρ ≤ 2ρ˜ for all (x, t) ∈ R3 × [0, T ], A (T ) ≤ 2ε 2 (1 + T )−3( p1 − 12 ) , 1 ⎩A (T ) ≤ 2ε, A (T ) + A (T ) ≤ 2ε 1/2 , A (σ (T )) ≤ 2K, 2 3 4 5 then the following estimates hold ⎧ ⎨0 ≤ ρ ≤ ρ ≤ 7 ρ˜ for all (x, t) ∈ R3 × [0, T ], A (T ) ≤ ε 2 (1 + T )−3( p1 − 12 ) , 1 4 ⎩A (T ) ≤ 3 ε, A (T ) + A (T ) ≤ ε 1/2 , A (σ (T )) ≤ K, 2 3 4 5 2

(4.6)

(4.7)

provided ε ≤ ε¯ . Throughout this section, we use the convention that C denotes a generic positive constant depending on μ, λ, ρ ∞ , a, γ , β, ρ, ˜ inf ,  H 3 , M1 and M2 , but not depending on the time x∈R3

T > 0. Remark 4.1. It is clear from (1.19) and (4.6) that

ρ − ρ∗ 2L2 ≤ Cε.

(4.8)

To deal with the standard energy estimate A2 (T ), we need the L2 -bound for η. Lemma 4.1. Under the conditions of Proposition 4.1, it holds that −3( p1 − 12 )

A1 (T ) ≤ ε 2 (1 + T )

(4.9)

,

provided ε ≤ ε¯ 1 . Proof. Similar to the proof of Lemma 3.5 in [7]. First, rewriting (1.3) into ηt − η = f := −∇ · (η(u − ∇)),

(4.10)

using the classical theory of linear parabolic equation, we get T η(t) = L(t)η0 +

L(T − τ )f (τ )dτ,

t ≥ 0,

0

where L(t) : φ → υ(·, t) is the solution semigroup defined by L(t) = e−t , i.e., υ(x, t) = L(t)φ = K(·, t) ∗ φ(·), where K := K(x, t) is the heat kernel

(4.11)

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

K(x, t) =

8675

|x|2 1 √ 3 e− 4t . (2 πt)

Noticing K(x, t) L1 = 1 and ∇ k (L(t)φ) = ∇ k K(t) ∗ φ, it follows from the Young inequality that − 32 ( p1 − 12 )− k2

∇ k K(t) ∗ φ L2 ≤ ∇ k K(t) Lr φ Lp ≤ C(1 + t) where

1 r

=1+

1 2

φ Lp ,

(4.12)

− p1 . Then, we deduce that t

η L2 ≤ L(t)η0 L2 +

L(t − τ )f (τ ) L2 dτ 0

− 32 ( p1 − 12 )

≤C(1 + t)

t

η0 Lp + C

∇K(t − τ ) ∗ (η(u − ∇)) L2 dτ 0

− 32 ( p1 − 12 )

≤C(1 + t)

t

η0 Lp + C

3

1

(1 + t − τ )−( 4 + 2 ) η(u − ∇) L1 dτ

0 − 32 ( p1 − 12 )

≤C(1 + t)

t +C

η0 Lp 5

(1 + t − τ )− 4 η L2 ( u L2 + ∇ L2 )dτ.

(4.13)

0

Next, we turn to get the bound of u L2 . Using the fact ρ∗ > 0, the Hölder inequality, Lemma 3.2, the Cauchy inequality and (4.8), we have   ρ∗ − ρ 2 1 2 ρ|u|2

u L2 = |u| + ρ∗ ρ∗ ≤C ρ∗ − ρ L2 u 2L4 + Cε 1

3

≤C ρ∗ − ρ L2 u L2 2 ∇u L2 2 + Cε 1 ≤ u 2L2 + C ∇u 4L2 + Cε, 2

(4.14)

which implies 1

u L2 ≤ C(ε 2 + ∇u 2L2 ).

(4.15)

Finally, by (2.1), (4.15) and (4.6), we infer from (4.13) that − 32 ( p1 − 12 )

η L2 ≤C(1 + t) 3

− 32 ( p1 − 12 )

≤C1 ε 2 (1 + t)

3

− 32 ( p1 − 12 )

η0 Lp + Cε 2 (1 + t) .

  Hence, choosing ε ≤ ε¯ 1  min 1, C1−2 , we complete the proof of (4.9). 2

(4.16)

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By Lemma 4.1, we can get the estimates of A2 (T ) and

T 0

∇η 2L2 dt.

Lemma 4.2. Under the conditions of Proposition 4.1, it holds that 3 A2 (T ) ≤ ε, 2

(4.17)

T sup η 2L2 +

0≤t≤T

∇η 2L2 dt ≤ Cε,

(4.18)

0

and T ( ∇u 4L2 + ∇η 4L2 )dt ≤ Cε,

(4.19)

0

provided ε ≤ ε¯ 2 . Proof. Using (1.2) and (1.8), we get ρut + ρu · ∇u + ρ(

∇pF ∇pF (ρ∗ ) ) + ∇η − μu − λ∇(∇ · u) = −η∇. − ρ ρ∗

(4.20)

Multiplying (4.20) by u, using (1.5), we obtain after integration by parts d dt



ρ|u|2 dx + 2

ρ

 ρu · ∇

pF (s) μ dsdx + ∇u 2L2 ≤ C η 2L2 , s 6

(4.21)

ρ∗

applying (1.1), we deduce ρ

 ρu · ∇

pF (s) dsdx = − s

ρ∗

=

ρ



d dt

∇ · (ρu) 

ρ∗

pF (s) dsdx = s

ρ

 ρt

pF (s) dsdx s

ρ∗

G(ρ)dx.

(4.22)

Therefore, putting (4.22) into (4.21), integrating it over [0, T ], we infer from (4.6) that    T ρ|u|2 G(ρ) + dx + ∇u 2L2 dt ≤ C2 ε 2 + ε, 2 0

 which, immediately leads to (4.17) provided that ε ≤ ε¯ 2  min ε¯ 1 , (2C2 )−1 .

(4.23)

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By (4.6), we deduce that T

σ(T )

∇u 4L2 dt

≤C

sup 0≤t≤σ (T )

0

≤Cε + Cε

∇u 2L2

T

∇u 2L2 dt

+C

0

3/2

sup σ (T )≤t≤T

σ ∇u 2L2

≤ Cε.

∇u 2L2 dt σ (T )

(4.24)

Multiplying (1.3) by η, integrating by parts, we get d

η 2L2 + ∇η 2L2 ≤ C( η 2L2 ∇u 4L2 + η 2L2 ), dt integrating it over [0, T ], we infer from (4.9) and (4.24) that T

η 2L2

+

∇η 2L2 dt ≤ C(ε 2 + η0 2L2 ) ≤ Cε. 0

Similar to the proof of (4.24), we also get Lemma 4.2. 2

T 0

∇η 4L2 dt ≤ Cε. We complete the proof of

Next, we denote F  ρ∗−1 [(μ + λ)(∇ · u) − (pF (ρ) − pF (ρ∗ )) − η].

(4.25)

As Lemma 3.4 in [15], we give the estimates for ∇F L2 and ∇u L6 to deal with the external potential forces. Lemma 4.3. Under the conditions of Proposition 4.1, furthermore assume that F = F (x, t) is the modified effective viscous flux defined in (4.25), then it holds that

∇F L2 + ∇(ρ∗−1 curlu) L2 ≤ C( ρ u

˙ L2 + ∇u L3 + η L3 + ρ − ρ∗ 2L6 ), (4.26)



∇u L6 ≤ C ρ u

˙ L2 + ∇u L2 + ρ − ρ∗ L6 + ρ − ρ∗ 2L6 + η L2 + ∇η L2 . (4.27) Proof. It follows from (1.8) that ρ∗−1 (∇(pF + η) + (η + βρ)∇) =ρ∗−1 [∇(pF − pF (ρ∗ ) + η) − (βρ∗ )−1 (η + β(ρ − ρ∗ ))∇pF (ρ∗ )] =∇[ρ∗−1 (pF − pF (ρ∗ ) + η)] + ρ∗−2 [pF − pF (ρ∗ ) + η − β −1 (η + β(ρ − ρ∗ ))pF (ρ∗ )]∇ρ∗ . (4.28) Thus, by virtue of (1.2), (1.8) and (4.28), we deduce

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ρ∗−1 ρ u˙ − ∇F + μcurl(ρ∗−1 curlu) = −[(λ + μ)(∇ · u)∇ρ∗−1 − μ∇ρ∗−1 × (curlu)] + [pF − pF (ρ∗ ) + η − β −1 (η + β(ρ − ρ∗ ))pF (ρ∗ )]∇ρ∗−1  G1 + G2 ,

(4.29)

which gives

F = ∇ · (ρ∗−1 ρ u˙ − G1 − G2 ),

(4.30)

and μ (ρ∗−1 curlu) =μ∇(∇ · (ρ∗−1 curlu)) − μcurlcurl(ρ∗−1 curlu) =μ∇(curlu · ∇ρ∗−1 ) + curl(ρ∗−1 ρ u˙ − G1 − G2 ).

(4.31)

By the definitions of G1 , G2 , it follows from (1.17) that

G1 L2 + G2 L2 ≤C( ∇ρ∗ L6 ∇u L3 + ∇ρ∗ L6 η L3 + ∇ρ∗ L6 (ρ − ρ∗ )2 L3 ) ≤C( ∇u L3 + η L3 + ρ − ρ∗ 2L6 ).

(4.32)

In view of (1.17), (4.32), applying the L2 -estimates for the elliptic equations (4.30)–(4.31), we deduce that

∇F L2 + ∇(ρ∗−1 curlu) L2 ≤C( ρ u

˙ L2 + G1 L2 + G2 L2 + ∇ρ∗ L6 ∇u L3 ) ≤C( ρ u

˙ L2 + ∇u L3 + η L3 + ρ − ρ∗ 2L6 ). On the other hand, by (1.17), (4.25), (3.3), (4.26) and the standard Lp -estimates for the elliptic equation, we obtain

∇u L6 ≤C( F L6 + ρ − ρ∗ L6 + η L6 + ρ∗−1 curlu L6 ) ≤C( ∇F L2 + ρ − ρ∗ L6 + ∇η L2 + ∇(ρ∗−1 curlu) L2 ) ≤C( ρ u

˙ L2 + ∇u L2 + η L2 + ∇η L2 + ρ − ρ∗ 2L6 + ρ − ρ∗ L6 ) 1 + ∇u L6 , 2 which implies (4.27). Therefore, the proof of Lemma 4.3 is complete.

2

By Lemmas 4.1–4.3, we can derive preliminary bounds for A3 (T ) and A4 (T ).

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Lemma 4.4. t A3 (T ) ≤ Cε + C

σ ∇u 3L3 ds,

(4.33)

0

t A4 (T ) ≤ Cε + CA3 (T ) + C

σ 3 ∇u 4L4 ds,

(4.34)

0

provided ε ≤ ε¯ 2 . Proof. The proof of Lemma 4.4 is due to [12,13,15]. Multiplying the equation (1.3) by σ m (ηt + η) and integrating the result equation with respect to x over R3 , we have d m (σ ∇η 2L2 ) + σ m ( ηt 2L2 + ∇ 2 η 2L2 ) dt  ≤C σ m |∇ · (η(u − ∇))|2 dx + mσ m−1 σ  ∇η 2L2 ≤Cσ m ( η 2L∞ ∇u 2L2 + u 2L6 ∇η 2L3 + ∇η 2L2 ∇ 2H 2 + ∇η L2 ∇ 2 η L2 ∇ 2H 2 ) + mσ m−1 σ  ∇η 2L2 1 ≤ σ m ∇ 2 η 2L2 + Cσ m ( ∇η 2L2 ∇u 4L2 + ∇η 2L2 ) + mσ m−1 σ  ∇η 2L2 , 2 that is d m (σ ∇η 2L2 ) + σ m ( ηt 2L2 + ∇ 2 η 2L2 ) dt ≤Cσ m ∇η 2L2 ∇u 4L2 + C(mσ m−1 σ  + σ m ) ∇η 2L2 .

(4.35)

Choosing m = 1 in (4.35), integrating it over [0, T ], using Gronwall’s inequality, Lemma 4.2 and (4.6), we find that t sup 0≤t≤T

(σ ∇η(t) 2L2 ) +

σ ( ηt (s) 2L2 + ∇ 2 η(s) 2L2 )ds ≤ Cε.

(4.36)

0

It follows from (1.2) and (1.8) that ρ u˙ + ∇(pF − pF (ρ∗ ) + η) − μu − λ∇(∇ · u) = −(η + β(ρ − ρ∗ ))∇.

(4.37)

Multiplying (4.37) by σ m u˙ and integrating by part over R3 , we get    σ m ρ|u| ˙ 2 dx = (μσ m u · u˙ + λσ m ∇divu · u)dx ˙ − σ m (η + β(ρ − ρ∗ ))∇ · udx ˙ 

 −

σ u˙ · ∇(pF − pF (ρ∗ ))dx − m

σ u˙ · ∇ηdx  m

4  i=1

Ii .

(4.38)

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Now, we bind the terms I1 , I2 , I3 , I4 . Integration by parts gives that (also see (3.11)–(3.12) in [13]) μ λ I1 ≤ − ( σ m ∇u 2L2 + σ m ∇ · u 2L2 )t + Cmσ m−1 σ  ∇u 2L2 + Cσ m ∇u 3L3 . 2 2

(4.39)

Due to (1.1), we have ρt + ∇ · ((ρ − ρ∗ )u) + ∇ · (ρ∗ u) = 0.

(4.40)

By (1.17), (3.3) and (4.8), we derive that I2 ≤ −

d dt

 σ m (η + β(ρ − ρ∗ ))∇ · udx

+ Cmσ m−1 σ  ( ρ − ρ∗ L2 + η L2 ) u L6 ∇ L3 + Cσ m ∇u 2L2  + Cσ m (|u||∇u||∇| + |∇ρ∗ ||u|2 |∇| + |ρ − ρ∗ ||u|2 |∇ 2 |)dx + Cσ m ηt L2 u L6 ∇ L3  d σ m (η + β(ρ − ρ∗ ))u · ∇dx + C ∇u 2L2 + Cm2 σ 2m−1 σ  ε + Cσ m ηt 2L2 . ≤− dt (4.41) To deal with I3 , we observe from (1.1) and (1.4) that (pF )t + ∇ · (pF u) + (ρpF − pF )(∇ · u) = 0,

(4.42)

the inequality (3.16) in [15] gives that I3 ≤

d dt



σ m (∇ · u)(pF − pF (ρ∗ ))dx + C ∇u 2L2 + Cm2 σ 2m−1 σ  ε.

(4.43)

And it follows from the direct calculation and integration by parts that  I4 =

 σ m (∇ · ut )ηdx −

d ≤ dt



σ m (u · ∇u) · ∇ηdx

σ m (∇ · u)ηdx + Cmσ m−1 σ  ( ∇u 2L2 + η 2L2 ) + Cσ m ( ∇u 2L2 + ηt 2L2 )

+ Cσ m u L6 ∇u L3 ∇η L2  d σ m (∇ · u)ηdx + C( ∇u 2L2 + η 2L2 ) + Cσ m ηt 2L2 ≤ dt + Cσ m ∇η 6L2 + Cσ m ∇u 3L3 .

(4.44)

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Substituting (4.39) and (4.41)–(4.44) into (4.38), we have 1 λ μ ˙ 2L2 ( σ m ∇u 2L2 + σ m ∇ · u 2L2 )t + σ m ρ 2 u

2 2 d ≤ I0 + C( ∇u 2L2 + η 2L2 ) + Cm2 σ 2m−1 σ  ε dt

+ Cσ m ηt 2L2 + Cσ m ∇η 6L2 + Cσ m ∇u 3L3 ,

(4.45)

here  |I0 |  | ≤

σ m ((∇ · u)(pF − pF (ρ∗ )) − (η + β(ρ − ρ∗ ))u · ∇ + (∇ · u)η) dx|

μ m σ ∇u 2L2 + Cε. 12

Choosing m = 1 in (4.45), integrating it over (0, T ), using (1.5), (4.17), (4.19) and (4.36), we get (4.33). Operating σ m u˙ j [∂/∂t + div(u·)] to (4.37)j with m > 0, summing with respect to j , and integrating by parts over R3 , we obtain 

    σm m j 2 ρ|u| ˙ dx − σ m−1 σ  ρ|u| ˙ 2 dx − μ σ m u˙ j [ ut + div(u uj )]dx Ki  2 2 t i=1  − λ σ m u˙ j [∂t ∂j divu + div(u∂j divu)]dx

4 

 σ m u˙ j [∂j (pF )t + div(u · ∂j (pF − pF (ρ∗ )))]dx

=−  −

σ m u˙ j [∂j ηt + div(u · ∂j η)]dx  σ m u˙ j [ηt ∂j  + ∂k (uk · (η∂j ))]dx

−  − 

4 

σ m u˙ j [βρt ∂j  + β∂k (uk · ((ρ − ρ∗ )∂j ))]dx IIi .

(4.46)

i=1

By integrating by parts, the Hölder inequality, the Cauchy inequality, (1.17), (3.3), (4.8) and (4.42), we have  II1 = − ≤



σ m ∂j u˙ j ((γ − 1)pF (∇ · u) + ∇ · (pF (ρ∗ )u)) + ∂k u˙ j ∂j uk (pF − pF (ρ∗ )) dx

μ m ˙ 2L2 + Cσ m ∇u 2L2 , σ ∇ u

48

(4.47)

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II2 ≤

μ m ˙ 2L2 + Cσ m ηt 2L2 + Cσ m ∇u 4L2 + Cσ m ∇η 2L2 ∇ 2 η 2L2 , σ ∇ u

48

(4.48)

μ m ˙ 2L2 + Cσ m ηt 2L2 + Cσ m ∇u 4L2 + Cσ m ∇η 4L2 . σ ∇ u

48

(4.49)

and II3 ≤

Similarly, by virtue of (1.1), we obtain II4 ≤

1 μ m ˙ 2L2 + Cσ m ( ρ 2 u

˙ 2L2 + ∇u 2L2 ). σ ∇ u

48

(4.50)

As the estimates of N2 and N3 in [13], we deal with K3 , K4 as follows:  ˙ 2 + ∂i u˙ j ∂k uk ∂i uj − ∂i u˙ j ∂i uk ∂k uj − ∂i uj ∂i uk ∂k u˙ j ]dx K3 =μ σ m [|∇ u| ≥

11μ m ˙ 2L2 − Cσ m ∇u 4L4 , σ ∇ u

12

K4 ≥ (λ +

2μ m 9μ m ˙ 2L2 − ˙ 2L2 − Cσ m ∇u 4L4 . )σ ∇ · u

σ ∇ u

3 12

(4.51) (4.52)

Substituting (4.47)–(4.52) into (4.46), we arrive at 1

˙ 2L2 )t + σ m ∇ u

˙ 2L2 (σ m ρ 2 u

≤Cσ m ( ∇u 2L2 + ηt 2L2 + ∇η 4L2 + ∇u 4L2 + ∇η 2L2 ∇ 2 η 2L2 ) 1

˙ 2L2 + Cσ m ∇u 4L4 . + C(mσ m−1 σ  + σ m ) ρ 2 u

(4.53)

Differentiating (1.3) with respect to t gives ηtt + ∇ · (ηt u + ηut − ηt ∇) − ηt = 0.

(4.54)

Multiplying (4.54) by σ m ηt , integrating by parts over R3 and using the fact ut = u˙ − u · ∇u, we get 1 d m m (σ ηt 2L2 ) + σ m ∇ηt 2L2 − σ m−1 σ  ηt 2L2 2 dt 2   = − σ m (ηt (∇ · u) + ∇ηt u) ηt dx − σ m (∇η · u˙ + η(∇ · u)) ˙ ηt dx   + σ m (∇η · (u · ∇u) + η(∇ · (u · ∇u)))ηt dx − σ m ηt ∇ · ∇ηt dx 

4 

IIIi .

i=1

Using the Hölder inequality, the Cauchy inequality, (3.3) and (4.27), we deduce that

(4.55)

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σm (4.56)

∇ηt 2L2 + Cσ m ηt 2L2 ∇u 4L2 , 8 σm σm III2 ≤Cσ m ηt L3 ∇ u

˙ L2 ∇η L2 ≤

∇ u

˙ 2L2 +

∇ηt 2L2 + Cσ m ∇η 4L2 ηt 2L2 , 4 8 (4.57) III1 ≤

III3 ≤Cσ m ∇ηt L2 ∇u L2 ∇η L2 ∇u L6   ˙ L2 + ∇u L2 + ∇η L2 + 1 ≤Cσ m ∇ηt L2 ∇u L2 ∇η L2 ρ u



σm

∇ηt 2L2 + Cσ m ∇η 2L2 ∇u 2L2 ρ u

˙ 2L2 + ∇u 2L2 + ∇η 2L2 + 1 , (4.58) ≤ 8 and III4 ≤

σm

∇ηt 2L2 + Cσ m ηt 2L2 . 8

(4.59)

Putting (4.56)–(4.59) into (4.55), we obtain (σ m ηt 2L2 )t + σ m ∇ηt 2L2

 σm ≤C mσ m−1 σ  + σ m ηt 2L2 + Cσ m ηt 2L2 ∇u 4L2 +

∇ u

˙ 2L2 + Cσ m ∇η 4L2 ηt 2L2 2 

+ Cσ m ∇η 2L2 ∇u 2L2 ρ u

(4.60) ˙ 2L2 + ∇u 2L2 + ∇η 2L2 + 1 . Finally, summing up (4.53) and (4.60), choosing m = 3 and then integrating the results over (0, T ), one gets sup σ

3



ρ

0≤t≤T

T ≤C

1 2

u

˙ 2L2

+ ηt 2L2



T +

 σ 3 ∇ u

˙ 2L2 + ∇ηt 2L2 dt

0 2 



(σ σ + σ ) ρ 3

1 2

u

˙ 2L2

+ ηt 2L2



T dt + C

0

 σ 3 ∇u 2L2 + ∇η 2L2 + ηt 2L2 dt

0

T +C

σ

3



∇u 4L2

+ ∇η 4L2



T dt + C

0

0

+C

+C

σ 3 ∇ 2 η 2L2 ∇η 2L2 dt 0

T σ 3 ∇u 4L2 ηt 2L2 dt + C

T

σ

∇u 4L4 dt

0

T +C

T 3

σ 3 ∇η 4L2 ηt 2L2 dt 0

1

σ 3 ∇η 2L2 ∇u 2L2 ρ 2 u

˙ 2L2 dt 0

T +C 0

9

  σ 3 ∇u 2L2 ∇η 2L2 ∇u 2L2 + ∇η 2L2 dt  IVi . i=1

(4.61)

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Using (4.6), (4.17), (4.18) and (4.36), it holds that IV1 + IV5 ≤ CA3 (T ), and 1

4

3

IV2 + IV3 + IV6 + IV7 + IV8 + IV9 ≤ Cε + Cε 2 + 5 + Cε 2 ≤ Cε. Inserting the above estimates into (4.61), leads to

sup σ

3



ρ

1 2

0≤t≤T

u

˙ 2L2

+ ηt 2L2



T +

 σ 3 ∇ u

˙ 2L2 + ∇ηt 2L2 dt

0

T ≤Cε + CA3 (T ) + C

σ 3 ∇u 4L4 dt.

(4.62)

0

In addition, it follows from (1.3) that

∇ 2 η L2 ≤ C( ηt L2 + ∇η L3 u L6 + η L∞ ∇u L2 + ∇η L3 ∇ L6 + η L∞  L2 ) 1

1

1

1

≤ C( ηt L2 + ∇η L2 2 ∇ 2 η L2 2 ∇u L2 + ∇η L2 2 ∇ 2 η L2 2 ∇ H 1 ),

(4.63)

which, together with (4.62) and (4.6), gives that

sup 0≤t≤T



 σ 3 ∇ 2 η 2L2 ≤ C sup σ 3 ηt 2L2 0≤t≤T

 + C sup σ 3 ∇η 2L2 ∇u 4L2 + ∇η 2L2 ∇ 4H 1 0≤t≤T

T ≤ Cε + CA3 (T ) + C

σ 3 ∇u 4L4 dt 0



2 + C sup σ ∇η 2L2 sup σ ∇u 2L2 0≤t≤T

0≤t≤T

T ≤ Cε + CA3 (T ) + C

σ 3 ∇u 4L4 dt. 0

Hence, we finish the proof of (4.34). 2 The next lemma plays an important role in the proofs of A3(T ) and A4 (T ).

(4.64)

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Lemma 4.5. Under the conditions of Proposition 4.1, it holds that T

 σ 3 ∇u 4L4 + ρ − ρ∗ 4L4 + F 4L4 + η 4L4 dt ≤ Cε,

(4.65)

0

provided ε ≤ ε¯ 3 . Proof. In terms of the effective viscous flux F in (4.25), we can rewrite (1.1) as (ρ − ρ∗ )t +

ρ∗ ρ2F ρ∗ η (pF (ρ) − pF (ρ∗ )) = −∇ · (u(ρ − ρ∗ )) − u · ∇ρ∗ − ∗ − . μ+λ μ+λ μ+λ

(4.66)

Multiplying (4.66) by 4(ρ − ρ∗ )3 and then integrating it over R3 , we have   d 4 (ρ − ρ∗ )4 dx + ρ∗ (pF (ρ) − pF (ρ∗ ))(ρ − ρ∗ )3 dx dt μ+λ 

 ≤C (ρ − ρ∗ )4 |∇u| + |ρ − ρ∗ |3 |u||∇ρ∗ | + |ρ − ρ∗ |3 |F | + |ρ − ρ∗ |3 |η| dx ≤ ρ − ρ∗ 4L4 + C( ∇u 2L2 + F 4L4 + η 4L4 ),

(4.67)

where we have used (1.17), (1.18) and the Cauchy–Schwarz inequality with  > 0. By (1.17), there exists a positive constant C such that ρ∗ (pF (ρ) − pF (ρ∗ ))(ρ − ρ∗ )3 ≥ C(ρ − ρ∗ )4 .

(4.68)

Summing up (4.68) and (4.67), choosing  > 0 small enough, multiplying the result by σ 3 and integrating it over (0, T ), we get T

T σ ρ 3

− ρ∗ 4L4 dt

≤ Cε + C

0

σ 3 F 4L4 dt,

(4.69)

0

where we have used (4.8), (4.17) and T

T σ

0

3

η 4L4 dt

≤C

T σ

3

η 2L2 ∇η 2L2 dt

0

+C

σ 3 ∇η 4L2 dt ≤ Cε.

(4.70)

0

It follows from (4.25) and (1.17) that

∇u L4 ≤ C( ∇ · u L4 + curlu L4 ) ≤ C( F L4 + ρ − ρ∗ L4 + η L4 + ρ∗−1 curlu L4 ). (4.71) Using (4.69)–(4.71), (3.3), (4.25), (4.6), (4.26) and (4.18), we obtain

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T

 σ 3 ∇u 4L4 + ρ − ρ∗ 4L4 + F 4L4 dt

0

T ≤Cε + C

 σ 3 F 4L4 + ρ∗−1 curlu 4L4 dt

0

T ≤Cε + C

 σ 3 F L2 ∇F 3L2 + ρ∗−1 curlu L2 ∇(ρ∗−1 curlu) 3L2 dt

0

T ≤Cε + C



 1 σ 3 ∇u L2 + ε 2 ρ u

˙ 3L2 + ∇u 3L3 + η 3L3 + ρ − ρ∗ 6L6 dt

0

T



1

 1 1 2 ≤Cε + C ˙ 2L2 σ 2 ∇u L2 + ε 2 σ 3 ρ u

σ ρ u

˙ 2L2 dt 0

T +C



 1 σ 3 ∇u L2 + ε 2 η 3L2 + ∇η 3L2 dt

0

T +C

∇u L2



1

σ 2 ∇u L2

 5   1 σ 2 ∇u 2L4 + σ 3 ε 2 ρ − ρ∗ 2L4 dt

0 1

T

+ Cε 2

 σ 3 ∇u 2L2 + ∇u 4L4 + ρ − ρ∗ 4L4 dt

0

≤Cε + C3 ε

1 2

T

 (σ 3 ∇u 4L4 + ρ − ρ∗ 4L4 dt,

(4.72)

0

where we have used 0 ≤ ρ ≤ 2ρ, ¯ 0 < ρ ≤ inf ρ∗ ≤ sup ρ∗ ≤ ρ¯ < ∞, (4.8) and the following inequality 1

ρ − ρ∗ 6L6 ≤ C ρ − ρ∗ 3L3 ≤ Cε 2 ρ − ρ∗ 4L4 .

(4.73)

 Choosing ε ≤ ε¯ 3  min ε¯ 2 , (2C3 )−2 , one immediately obtains (4.65) from (4.72) and (4.70). 2 We derive a bound for A5 (σ (T )):

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Lemma 4.6. Under the conditions of Proposition 4.1, it holds that σ(T )



A5 (σ (T )) +

 1 ˙ 2L2 + ηt 2L2 + ∇ 2 η 2L2 dt ≤ K,

ρ 2 u

(4.74)

0

provided ε ≤ ε¯ 4 . Proof. Taking m = 0 in (4.35) and (4.45), putting them together and integrating the results over [0, σ (T )], yields σ(T )



A5 (σ (T )) + 2

 1 ˙ 2L2 + ηt 2L2 + ∇ 2 η 2L2 dt

ρ 2 u

0 σ(T )

≤C(ε + M1 + M2 ) + C

3 2

σ(T )

3 2

∇u L2 ∇u L6 dt + C

( ∇η 2L2 ∇u 4L2 + ∇η 6L2 )dt

0

≤

K +C 2

σ(T )

∇u 6L2 dt + 0

≤

0 σ(T )

σ(T )

1

ρ 2 u

˙ 2L2 dt + C

( ∇η 2L2 ∇u 4L2 + ∇η 6L2 )dt

0

K + C4 εA25 (σ (T )) + 2

0

σ(T )

1

ρ 2 u

˙ 2L2 dt.

(4.75)

0

 Hence, (4.74) is proved provided that ε ≤ ε¯ 4  min ε¯ 3 , (8KC4 )−1 .

2

We close the estimates of A3 (T ) and A4 (T ). Lemma 4.7. Under the conditions of Proposition 4.1, it holds that A3 (T ) + A4 (T ) ≤ ε 1/2 ,

(4.76)

provided ε ≤ ε¯ 5 . Proof. Putting (4.33) and (4.34) together, using the Hölder inequality, (4.65) and (4.17), we have σ(T )

A3 (T ) + A4 (T ) ≤Cε + C

T

σ ∇u 3L3 dt

+C

0

σ (T )

σ(T )

≤Cε + C

σ ∇u 3L3 dt

3 2

3 2

T

σ ∇u L2 ∇u L6 dt + C 0

σ (T )

σ(T )

≤Cε + C

3

3

σ ∇u L2 2 ∇u L2 6 dt. 0

( ∇u 2L2 + σ 3 ∇u 4L4 )dt

(4.77)

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As a consequence of (4.27), (4.6) and (4.17), one obtains

2 2 3 ˙ L2 + ∇u L2 + ρ − ρ∗ L6 + ρ − ρ∗ 2L6 + η L2 + ∇η L2 σ ∇u L3 6 ≤ Cσ ρ u

1

≤ Cε 9 .

(4.78)

Collecting (4.77), (4.78), (4.6), (4.27), (4.65), (4.73) and (4.74) shows that A3 (T ) + A4 (T ) ≤Cε + Cε

σ(T )

1 9

7

5

∇u L6 2 ∇u L6 6 dt 0 σ(T )

7

∇u L6 2 ρ u

˙ L2 + ∇u L2 + ρ − ρ∗ L6 + ρ − ρ∗ 2L6

1

≤Cε + Cε 9 0

+ η L2 + ∇η L2 ≤Cε + Cε

25 36

5

6

dt

⎞ 125 ⎛ σ (T )  ⎝ ( ρ u

˙ 2L2 + ∇u 2L2 + ρ − ρ∗ 2L6 + ρ − ρ∗ 4L6 + η 2L2 + ∇η 2L2 )dt ⎠ 0

≤C5 ε

25 36

1 2

≤ε ,

  − 36 provided that ε ≤ ε¯ 5 = min ε¯ 4 , C5 7 . Hence, (4.76) is proved.

2

Lemma 4.8. Under the conditions of Proposition 4.1, furthermore assume that K > 0 satisfies the one in Lemma 4.6. Then there holds that T sup

t∈[0,T ]

( ∇u 2L2

+ ∇η 2L2 ) +

1

( ρ 2 u

˙ 2L2 + ηt 2L2 + ∇ 2 η 2L2 )dt ≤ C,

(4.79)

0

and T sup

t∈[0,T ]

σ ρ u

˙ 2L2

+

σ ∇ u

˙ 2L2 dt ≤ C,

(4.80)

0

provided ε ≤ ε¯ 5 . Proof. The combination of (4.74) and (4.76) immediately gives (4.79). To prove (4.80), choosing m = 1 in (4.53), integrating it over (0, T ), using (4.9), (4.17), (4.18), (4.8), (4.27), (4.65) and (4.79), yields that

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

sup (σ ρ

1 2

t∈[0,T ]

T u

˙ 2L2 ) +

σ ∇ u

˙ 2L2 dt 0

⎛ ≤C ⎝ε +

T

8689

T

1 2

ρ u

˙ 2L2 dt + 0

⎞ σ ∇u 4L4 dt ⎠

0

σ(T )

σ ∇u L2 ∇u 3L6 dt

≤C + C 0

σ(T )

 σ ρ u

˙ 3L2 + ∇u 3L2 + ρ − ρ∗ 3L6 + ρ − ρ∗ 6L6 + η 3L2 + ∇η 3L2 dt

≤C + C 0

1

1

˙ L2 ), ≤C + C sup (σ 2 ρ 2 u

t∈[0,T ]

which implies (4.80) in view of Young’s inequality. 2 Finally, the application of Lemma 3.3 gives the uniform upper bound of the density ρ, which will be the key to obtain all the higher–order estimates and extend the classical solution globally. Lemma 4.9. Under the conditions of Proposition 4.1, it holds that 7 ˜ sup ρ L∞ ≤ ρ, 4 0≤t≤T  provided ε ≤ ε¯  min ε¯ 7 ,



ρ˜ 4C

2 

(4.81)

.

Proof. Let Dt  ∂t + u · ∇. Then we can rewrite (1.1) as Dt ρ = g(ρ) + b (t) where aρ γ g(ρ)  − (ρ γ − ρ∗ ), μ+λ

1 b(t)  − μ+λ

t

ρ(F˜∗ + η)ds,

0

and F˜∗  (μ + λ)(∇ · u) − (pF − pF (ρ∗ )) − η.

(4.82)

It follows from the definition of g that lim g(ρ) = −∞. Thus, we want to estimate b(t). ρ→∞

First, we use (1.2) and (1.8) to get that

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F˜∗ = ∇ · (ρ u˙ + (η + β(ρ − ρ∗ ))∇) .

(4.83)

Applying Lp -estimate of the elliptic system for (4.83), using (1.17), (4.9), (4.8) and (4.80), we deduce that 1

3

∇ F˜∗ L4 ≤C ρ u

˙ L4 2 ρ u

˙ L4 6 + C η L4 ∇ L∞ + C ρ − ρ∗ L12 ∇ L6 3

1

1

≤Cσ − 8 ∇ u

˙ L4 2 + Cε 12 + C ∇η 2L2 .

(4.84)

In view of (3.4), (4.9), (4.8), (4.79) and (4.84), we have 1

6

F˜∗ L∞ ≤C F˜∗ L7 2 ∇ F˜∗ L7 4 1

1

1

3

1

1

6

≤C( ∇u L7 2 + η L7 2 + ρ − ρ∗ L7 2 )(σ − 8 ∇ u

˙ L4 2 + ε 12 + ∇η 2L2 ) 7 9

3

12

1

≤Cσ − 28 ∇ u

˙ L142 + Cε 14 + C ∇η L72 .

(4.85)

For 0 ≤ t1 < t2 ≤ σ (T ) ≤ 1, we infer from (4.85), (4.76), (4.79) and (4.80) that σ(T )

( F˜∗ L∞ + η L∞ )dt

|b(t2 ) − b(t1 )| ≤C 0

σ(T )

1

4



σ − 7 σ ∇ u

˙ 2L2

≤Cε 14 + C

1

4

˙ 2L2 σ 3 ∇ u

1

14

⎛ σ (T ) ⎞ 67  dt + C ⎝

∇η 2L2 dt ⎠

0

≤Cε

1 14

0

+ C (A4 (T ))

1 14

1 28

≤ Cε .

(4.86)

Thus, for t ∈ [0, σ (T )], N0 , N1 and ζ in Lemma 3.3 can be chosen as follows: 1

N0 = Cε 28 ,

N1 = 0,

ζ = ρ. ¯

It holds that 0 < ρ ≤ ρ∗ ≤ ρ¯ < ∞ in (1.17) and g(ξ ) = −

 aξ  γ ξ − ρ¯ γ ≤ −N1 = 0, μ+λ

∀ ξ ≥ ζ = ρ, ¯

therefore, it follows from (3.6) that 1 3

ρ L∞ ≤ max{ρ, ¯ ρ} ˜ + N0 ≤ ρ˜ + Cε 28 ≤ ρ, ˜ 2 0≤t≤σ (T )

sup

where 



ρ˜ ε ≤ ε¯ 6  min ε¯ 5 , 2C

28  .

(4.87)

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8691

On the other hand, for σ (T ) ≤ t1 < t2 ≤ T , we use (4.8), (4.9), (4.17), (4.84)1 , (4.79), (4.76) and (4.80) to get that t2 |b(t2 ) − b(t1 )| ≤C

( F˜∗ L∞ + η L∞ )dt ≤ C

t1

t2

1

6

1

1

( F˜∗ L7 2 ∇ F˜∗ L7 4 + ∇η L2 2 ∇ 2 η L2 2 )dt

t1

6 t2  1 3 7 4 4 ˙ L6 + η L4 ∇ L∞ + ρ − ρ∗ L12 ∇ L6 dt

ρ u

˙ L2 ρ u

≤C t1

t2 σ ∇ 2 η 2L2 dt +

+C t1

t2

1

3

a (t2 − t1 ) 4(μ + λ)

9

1

1

ρ 2 u

˙ L142 ∇ u

˙ L142 dt + Cε 14 (t2 − t1 ) + Cε 2 +

≤C t1

T ≤C

3

σ ( ρ

1 2

 u

˙ 2L2

+ ∇ u

˙ 2L2 )dt

+ Cε

1 14

σ (T ) 1

≤Cε 2 +

a (t2 − t1 ) 4(μ + λ)

 1 a + (t2 − t1 ) + Cε 2 2(μ + λ)

a (t2 − t1 ), μ+λ

(4.88)

where ε in the last inequality of (4.88) satisfying 



a ε ≤ ε¯ 7  min ε¯ 6 , 2C(μ + λ)

14  .

Thus, for t ∈ [σ (T ), T ], N0 , N1 and ζ in Lemma 3.3 can be chosen as follows: N0 = Cε 1/2 ,

N1 =

a , μ+λ

ζ = ρ¯ + 1.

It holds that g(ξ ) = −

 aξ  γ a ξ − ρ¯ γ ≤ −N1 = − , μ+λ μ+λ

∀ ξ ≥ ζ = ρ¯ + 1,

therefore, it follows from Lemma 3.3 and (4.87) that 

sup σ (T )≤t≤T

where

ρ L∞

 3ρ˜ 3ρ˜ 7 ≤ max , ρ¯ + 1 + N0 ≤ + Cε1/2 ≤ ρ, ¯ 2 2 4

(4.89)

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ρ˜ ε ≤ ε¯  min ε¯ 7 , 4C

2  .

Hence, the combination of (4.87) and (4.89) yields (4.81).

2

Now, we can give the proof of Proposition 4.1. Proof of Proposition 4.1. By virtue of Lemmas 4.1, 4.2, 4.6–4.7, 4.9, the proof of Proposition 4.1 is complete. 2 5. Time-dependent higher-order estimates of (1.1)–(1.8) In this section, we proceed to prove the higher-order estimates of (ρ, u, η) to (1.1)–(1.8), which guarantee the existence of global classical solutions. To this end, we assume that the initial energy C0 satisfies (1.15), the conditions of Theorem 2.1 hold and C which will be used to denote the various positive constants may depend on 1

ρ0 − ρ ∞ H 3 , pF (ρ0 ) − pF (ρ ∞ ) H 3 , ρ 2 g L2 , ∇g L2 , ∇u0 H 2 , ∇η0 H 2 , T , ˜ β, M1 , M2 , inf  and ∇ H 3 , where g is the function in the combesides μ, λ, a, γ , ρ ∞ , ρ, x∈R3

patibility condition (2.2). Lemma 5.1. The following estimates hold

1  T

 2 2 2 2 ˙ L2 + ∇ η L2 + ηt L2 + sup ρ 2 u

∇ u

˙ 2L2 + ∇ηt 2L2 dt ≤ C,

0≤t≤T

(5.1)

0

sup 0≤t≤T





T

∇ρ L2 ∩L6 + ∇u H 1 +

∇u L∞ dt ≤ C.

(5.2)

0

Proof. Taking m = 0 in (4.53) and (4.60), putting the two equations into (4.63), integrating the results over (0, T ), using (4.76), (4.17), (4.18), (4.79) and the compatibility condition (2.2), we deduce that

 T

 1/2 2 2 2 2 sup ρ u

˙ L2 + ∇ η L2 + ηt L2 +

∇ u

˙ 2L2 + ∇ηt 2L2 dt

0≤t≤T

0

 2 2 ≤ (ρ 1/2 u)(0)

˙ +

η (0)

+C t L2 L2

T

∇u 4L4 dt 0

T

1

( ∇u 2L2 + ∇η 2L2 + ηt 2L2 + ∇ 2 η 2L2 + ρ 2 u

˙ 2L2 )dt

+C 0

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

8693

T ≤C + C

∇u L2 ∇u 3L6 dt 0

T ≤C + C

ρ 1/2 u

˙ 3L2 dt 0

˙ L2 , ≤C + C sup ρ 1/2 u

t∈[0,T ]

which, in view of Young’s inequality immediately leads to (5.1). For 2 ≤ r ≤ 6, we infer from (1.1) that (|∇ρ|r )t + ∇ · (|∇ρ|r u) + (r − 1)|∇ρ|r (∇ · u) + r|∇ρ|r−2 (∇ρ)t ∇u(∇ρ) + rρ|∇ρ|r−2 ∇ρ · ∇(∇ · u) = 0.

(5.3)

By direct calculations, we get d ˙ Lr ) ,

∇ρ Lr ≤ C ∇u L∞ ∇ρ Lr + C ∇ 2 u Lr ≤ C (1 + ∇u L∞ ) ∇ρ Lr + C (1 + ρ u

dt (5.4) where we have used the Lp -estimate that ˙ Lr + ∇ρ Lr + ∇η Lr + η Lr + 1) ≤ C ( ρ u

˙ Lr + ∇ρ Lr + 1) .

∇ 2 u Lr ≤ C ( ρ u

(5.5) In addition, by (3.3), (4.79) and (5.5), we infer from (3.8) with q˜ = 6 that  

∇u L∞ ≤C + C ( ∇ · u L∞ + ω L∞ ) ln e + ∇ u

˙ L2 + ∇ρ L6   ˙ L2 ≤C + C ( ∇ · u L∞ + ω L∞ ) ln e + ∇ u

  + C ( ∇ · u L∞ + ω L∞ ) ln e + ∇ρ L6 .

(5.6)

Since F˜ in (4.82) satisfies ρ u˙ − ∇ F˜ + μcurl(curlu) = − (η + β(ρ − ρ∗ )) ∇, thus, applying Lp -estimate for elliptic system, we arrive at     ˙ L2 + 1 . ˙ L6 + (η + β(ρ − ρ∗ )) ∇ L6 ≤ C ∇ u

∇ F˜ L6 + curl(curlu) L6 ≤C ρ u

(5.7) And we use (4.82), (3.4), (4.79), (5.1) and (5.7) to get that

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∇ · u L∞ + ω L∞ ≤C 1 + F˜ L∞ + η L∞ + ω L∞

 ≤C 1 + F˜ L2 + ω L2 + ∇ F˜ L6 + ∇ω L6   ≤C ∇ u

˙ L2 + 1 .

(5.8)

Setting     ˙ L2 , (t)  1 + ∇ u

˙ L2 ln e + ∇ u

(t)  e + ∇ρ L6 ,

then it follows from (5.4), (5.6) and (5.8) with r = 6 that  (t) ≤ C(t)(t) ln (t), which, gives d ln (t) ≤ C(t) ln (t), dt

(5.9)

due to the fact that (T ) ≥ e. By virtue of (5.1), we find T

T dt ≤ C + C

∇ u

˙ 2L2 dt ≤ C.

0

(5.10)

0

Applying Gronwall’s inequality for (5.9), together with (5.10), we deduce that sup ∇ρ(t) L6 ≤ sup (t) ≤ C.

0≤t≤T

(5.11)

0≤t≤T

Collecting (5.6), (5.8), (5.1) and (5.11), gives T

∇u L∞ dt ≤ C.

(5.12)

0

Taking r = 2 in (5.4), using (5.12), (4.79) and Gronwall’s inequality, we get sup ∇ρ(t) L2 ≤ C,

(5.13)

0≤t≤T

which, together with (5.5), (5.1), (5.11) and (5.12), shows (5.2).

2

Lemma 5.2. The following estimates hold sup ρ 0≤t≤T

sup

t∈[0,T ]



1 2

T ut 2L2

+

∇ut 2L2 dt ≤ C,

(5.14)

0



∇ρ H 1 + ∇pF H 1 +

T

0



∇ 2 u 2H 1 + ∇ 2 η 2H 1 dt ≤ C,

(5.15)

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

8695

and T sup ( ρt H 1 + pF t H 1 ) +

0≤t≤T

( ρtt 2L2 + pF tt L2 )2 dt ≤ C.

(5.16)

0

Proof. By ut = u˙ − u · ∇u, one easily gets (5.14) from the following simple calculations: 

 ρ|ut | dx ≤ 2

 ρ|u| ˙ dx + 2

ρ|u · ∇u|2 dx ≤ C + C ρ 1/2 u L2 u L6 ∇u 2L6 ≤ C,

and ˙ 2L2 + ∇(u · ∇u) 2L2

∇ut 2L2 ≤ ∇ u

≤ ∇ u

˙ 2L2 + C u 2L∞ ∇ 2 u 2L2 + C ∇u 4L4 ≤ ∇ u

˙ 2L2 + C, due to Lemmas 4.8, 5.1. Next, noticing that pF = aρ γ satisfies pF t + u · ∇pF + γ pF (∇ · u) = 0,

(5.17)

together with (1.1), (3.3) and (5.2), we get  d 2 2

∇ ρ L2 + ∇ 2 pF 2L2 dt



 ≤C ∇u L∞ ∇ 2 ρ 2L2 + ∇ 2 pF 2L2 + C ∇ 3 u L2 ∇ 2 ρ L2 + ∇ 2 pF L2   

+ C ∇ 2 u L6 ∇ρ L3 + ∇pF L3 ∇ 2 ρ L2 + ∇ 2 pF L2 

(5.18) ≤C (1 + ∇u L∞ ) ∇ 2 ρ 2L2 + ∇ 2 pF 2L2 + C ∇ 2 u 2H 1 + C. Using (5.2), (5.14), (4.79) and Lemma 5.1, we deduce from (1.2) and the standard L2 -estimate that   ˙ H 1 + ∇pF H 1 + ∇η H 1 + (η + βρ)∇ H 1 + 1

∇ 2 u H 1 ≤C ρ u

 ≤C 1 + ∇ut L2 + ∇ 2 pF L2 ,

(5.19)

also, using (5.1), (5.2), we deduce from (1.3) that  

∇ 2 η H 1 ≤ C 1 + ∇ut L2 + ∇ηt L2 ,

(5.20)

which, together with (5.1), (5.2), (5.14), (5.18), (5.19) and Gronwall inequality, immediately gives (5.15).

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Finally, we infer from (1.1) and (5.17) that

ρt H 1 + pF t H 1     ≤C u L∞ ∇ρ H 1 + ∇pF H 1 + C ∇u H 1 + C ∇u L6 ∇ρ L3 + ∇pF L3 

(5.21) ≤C 1 + ∇u 2H 1 + ∇ρ 2H 1 + ∇pF 2H 1 ≤ C, where we have used (5.2), (5.15) and Lemma 3.2. Moreover, (5.17) implies pF tt + ut · ∇pF + u · ∇pF t + γ pF t (∇ · u) + γ pF (∇ · ut ) = 0,

(5.22)

which, together with (5.2), (5.14), (5.15) and (5.21), gives T

T

pF tt 2L2 dt

0



ut L6 ∇pF L3 + u W 1,∞ pF t H 1 + ∇ut L2

≤C

2

dt

0

≤C + C

T



∇u 2H 2 + ∇ut 2L2 dt ≤ C.

(5.23)

0

Similarly, one also has ρtt L2 ∈ L2 (0, T ). Hence, combining this with (5.21) and (5.23) completes the proof of (5.16). 2 Lemma 5.3. It holds that

 sup ∇u 2H 2 + ∇ut 2L2 + ∇η 2H 2 + ∇ηt 2L2 0≤t≤T

+

T

 1

ρ 2 utt 2L2 + ∇ut 2H 1 + ηtt 2L2 + ∇ηt 2H 1 dt ≤ C.

(5.24)

0

Proof. Differentiating (1.2) with respect to t gives ρutt − μut − λ∇(∇ · ut ) = −ρt ut − (ρu · ∇u)t − ∇pF t − ∇ηt − (ηt + βρt )∇.

(5.25)

Multiplying (5.25) and (4.54) by utt and ηtt , respectively, and integrating the results over R3 , we obtain after adding them together that 1 d 2 dt  =−

 1  μ ∇ut 2L2 + λ ∇ · ut 2L2 + ∇ηt 2L2 + ρ 2 utt 2L2 + ηtt 2L2   ρt ut + ρt u · ∇u + ρut · ∇u + ρu · ∇ut + ∇pF t + ∇ηt + (ηt + βρt )∇ · utt dx

 −

(∇ · (η(u − ∇))t ) · ηtt dx

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

=

d dt + −

 − 1 2 



1 2



 ρt |ut |2 dx −

8697



 (ρt u · ∇u + (ηt + βρt )∇) · ut dx +

(pF t + ηt )(∇ · ut )dx



  (ρt u · ∇u)t · ut dx − ρut · ∇u · utt dx − (ηtt + βρtt )∇ · ut dx   ρu · ∇ut · utt dx − (pF tt + ηtt )∇ · ut dx − (∇ · (η(u − ∇))t )ηtt dx ρtt |ut |2 dx +

 d Vi . V0 + dt 7



(5.26)

i=1

Using (1.1) and integrating by parts, we bind the terms on the right-hand side of (5.26) from Lemmas 3.2, 5.1–5.2 and Cauchy–Schwarz inequality that    V0 = − ρu · (ut · ∇ut ) + (ρt u · ∇u + (ηt + βρt )∇) · ut − pF t (∇ · ut ) − ηt (∇ · ut ) dx ≤C u L∞ ρ 1/2 ut L2 ∇ut L2 + C ρt L2 u L∞ ∇u L3 ut L6 + C ηt L2 ∇ L3 ut L6 + C ρt L2 ∇ L3 ut L6 + C pF t L2 ∇ut L2 + C ηt L2 ∇ut L2 μ ≤ ∇ut 2L2 + C, 12 V 1 + V4 + V6   ≤C ρt L3 u L∞ ut L6 + ρut L3 ut L6 + ηtt L2 + ρtt L2 + pF tt L2 ∇ut L2

 1 1/2 1/2 ≤C ∇ut 2L2 + ρ 1/2 ut L2 ut L6 ∇ut 2L2 + ρtt 2L2 + pF tt 2L2 + ηtt 2L2 4

 1 ≤C 1 + ∇ut 4L2 + ρtt 2L2 + pF tt 2L2 + ηtt 2L2 , 4   1 V3 + V5 ≤ C ut L6 ∇u L3 + u L∞ ∇ut L2 ρ 1/2 utt L2 ≤ ρ 1/2 utt 2L2 + C ∇ut 2L2 , 2 and V2 + V7 ≤C( ρtt L2 u L∞ ∇u L3 ut L6 + ρt L2 ut 2L6 ∇u L6 + ρt L6 u L6 ∇ut L2 ut L6 ) + C( ∇ηt L3 u L6 + ut L6 ∇η L3 + η L∞ ∇ · ut L2 + ηt L3 ∇ · u L6 ) ηtt L2 + C( ∇ L∞ ∇ηt L2 +  L6 ηt L3 ) ηtt L2

 1 ≤C ρtt 2L2 + ∇ut 2L2 + ∇ηt 2L2 + 1 + ηtt 2L2 , 4 where we have used the following inequality  

∇ 2 ηt L2 ≤ C ηtt + ∇ · (η(u − ∇))t L2 ≤ C 1 + ηtt L2 + ∇ut L2 + ∇ηt L2 . (5.27)

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The regularity of the local solution, (3.2) gives T∗ /2

sup 0≤t≤T∗ /2

∇ut (·, T∗ /2) 2L2

+ ∇ηt (·, T∗ /2) 2L2 ) +



 ρ|utt |2 + |ηtt |2 dxdt ≤ C, (5.28)

0

and (t∇ut , t∇ηt ) ∈ C([0, T∗ ]; L2 ) which implies ( ∇ut (·, T∗ /2) L2 + ∇ηt (·, T∗ /2) L2 ) ≤

2 ( t∇ut L∞ (0,T∗ ;L2 ) + t∇ηt L∞ (0,T∗ ;L2 ) ) ≤ C, T∗ (5.29)

where C may depend on ∇g L2 . By virtue of the estimates of Vi (i = 0, 1, . . . , 7), integrating (5.26) over [0, T ], using Gronwall’s inequality, (5.14), (5.16) and (5.29), we have

sup T∗ /2≤t≤T

 T



∇ut 2L2 + ∇ηt 2L2 +

ρ 1/2 utt 2L2 + ηtt 2L2 dt ≤ C.

(5.30)

T∗ /2

Therefore, we obtain from (5.28) and (5.30) that

 T

 2 2 sup ∇ut L2 + ∇ηt L2 +

ρ 1/2 utt 2L2 + ηtt 2L2 dt ≤ C.

0≤t≤T

(5.31)

0

And we get from (5.19) and (5.20) that

sup 0≤t≤T



∇u 2H 2 + ∇η 2H 2 ≤ C.

(5.32)

Finally, we deduce from (5.25) that

 1

∇ 2 ut L2 ≤ C 1 + ρ 2 utt L2 + ∇ut L2 + ∇ηt L2 ,

(5.33)

which, together with (5.27) and (5.31), gives T



∇ 2 ut 2L2 + ∇ 2 ηt 2L2 dt ≤ C.

(5.34)

0

Hence, the combination of (5.31)–(5.32) and (5.34) finishes the proof of (5.24).

2

The following lemma is concerned with the H 3 -estimate of density ρ and pressure pF .

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

8699

Lemma 5.4. It holds that sup

t∈[0,T ]



∞



∞

ρ − ρ H 3 + pF − pF (ρ ) H 3 +

T



∇u 2H 3 + ∇η 2H 3 dt ≤ C.

(5.35)

0

Proof. Applying the standard L2 -estimate for elliptic system (1.2) again leads to

 ˙ L2 + ∇ 3 pF L2 + ∇ 3 η L2 + 1

∇ 4 u L2 ≤C ∇ 2 (ρ u)

 ≤C 1 + ∇ut H 1 + ∇ 3 pF L2 ,

(5.36)

where one has used Lemmas 5.1–5.3 and the following simple facts:



∇ 2 (ρut ) L2 ≤ C ∇ 2 ρ L2 ∇ut H 1 + ∇ρ L3 ∇ut L6 + ∇ 2 ut L2 ≤ C + C ∇ut H 1 , (5.37) and



∇ 2 (ρu · ∇u) L2 ≤ C 1 + ∇ 2 (ρu) L2 ∇u H 2 + ∇(ρu) L3 ∇ 2 u L6

 ≤ C 1 + ∇ 2 ρ L2 u L∞ + ∇ρ L6 ∇u L3 + ∇ 2 u L2 ≤ C. (5.38)

It follows from (5.17), (5.14) and (5.36) that 

∇ 3 pF 2L2 t

 ≤ C ∇ 3 u L2 ∇pF H 2 + ∇ 2 u L3 ∇ 2 pF L6 + ∇u L∞ ∇ 3 pF L2 ∇ 3 pF L2

 + C 1 + ∇ 2 ut L2 + ∇ 3 pF L2 ∇ 3 pF L2 ≤ C + C ∇ut 2H 1 + C ∇ 3 pF 2L2 ,

(5.39)

applying the Gronwall’s inequality for (5.39) and using (5.24), one obtains that sup ∇ 3 pF L2 ≤ C.

(5.40)

0≤t≤T

Combining the estimates of (1.19), (5.15), (5.24), (5.36) with (5.40) implies ∞

T

sup pF − pF (ρ ) H 3 +

0≤t≤T

∇u 2H 3 dt ≤ C.

(5.41)

0

By the similar arguments as (5.41), we get sup ρ − ρ ∞ H 3 ≤ C,

0≤t≤T

(5.42)

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and T

∇η H 3 dt ≤ C,

(5.43)

0

which, together with (5.41) and (5.42) gives (5.35).

2

Finally, we give the following initial-layer analysis. Lemma 5.5. For any given T > 0, it holds that

sup

τ ≤t≤T

+

∇ut 2H 1 + ∇ηt 2H 1 + ∇ 4 u 2L2 + ∇ 2 η 2H 2

T 



 |∇utt |2 + |∇ηtt |2 dxdt ≤ C(τ ).

(5.44)

τ

Proof. Differentiating (5.25) and (4.54) with respect to t , we get ρuttt + ρu · ∇utt − μutt − λ∇(∇ · utt ) = 2∇ · (ρu)utt + ∇ · (ρu)t ut − 2(ρu)t · ∇ut − (ρtt u + 2ρt ut ) · ∇u −ρutt · ∇u − ∇pF tt − ∇ηtt − (ηtt + βρtt )∇,

(5.45)

and ηttt − ηtt = −∇ · (η(u − ∇))tt .

(5.46)

Multiplying (5.45) and (5.46) by utt and ηtt , respectively, putting the results together and integrating them over R3 , we obtain 1 d 2 dt





  ρ|utt |2 + |ηtt |2 dx + μ|∇utt |2 + λ(∇ · utt )2 + |∇ηtt |2 dx 



= −4

ρu · ∇utt · utt dx −

(ρu)t · (∇(ut · utt ) + 2∇ut · utt ) dx 

 (ρtt u + 2ρt ut ) · ∇u · utt dx −

−  −

 ρutt · ∇u · utt dx +

(pF tt + ηtt )∇ · utt dx

 (ηtt + βρtt )∇ · utt dx −

∇ · (η(u − ∇))tt ηtt dx 

7 

VIi .

(5.47)

i=1

Next, we want to estimate the terms on the right-hand side of (5.47) (also see the results (3.95)–(3.98) in [13]). Using Lemmas 3.2, 5.1–5.4 and Cauchy–Schwarz inequality, we obtain

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VI1 ≤ u L∞ ρ 1/2 utt L2 ∇utt L2 ≤  ∇utt 2L2 + C() ρ 1/2 utt 2L2 ,    VI2 ≤C ρut L3 + u L∞ ρt L3 ∇ut L2 utt L6 + ut L6 ∇utt L2

 1/2 1/2 ≤C 1 + ρ 1/2 ut L2 ∇ut L2 ∇ut L2 ∇utt L2

 ≤ ∇utt 2L2 + C() 1 + ∇ut 3L2 ,   VI3 ≤C ρtt L2 u L∞ + ρt L3 ut L6 ∇u L3 utt L6

 ≤ ∇utt 2L2 + C() ρtt 2L2 + ∇ut 2L2 , VI4 ≤C ρ 1/2 utt L2 ∇u L3 utt L6 ≤  ∇utt 2L2 + C() ρ 1/2 utt 2L2 , VI5 ≤C( pF tt L2 + ηtt L2 ) ∇utt L2 ≤  ∇utt 2L2 + C()( pF tt 2L2 + ηtt 2L2 ),

   VI6 ≤C ηtt L2 + ρtt L2 ∇ L3 utt L6 ≤  ∇utt 2L2 + C() ρtt 2L2 + ηtt 2L2 , and   VI7 ≤C ∇ηtt L2 u L∞ + ∇η L3 utt L6 + ηtt L6 ∇u L3 ηtt L2 + C ∇ηt L2 ut L6 ηtt L3 + C ηtt L3 ∇ut L2 ηt L6   + C η L∞ ∇utt L2 + ∇ηtt L2 ∇ L∞ + ηtt L6 ∇ L3 ηtt L2

 ≤( ∇ηtt 2L2 + ∇utt 2L2 ) + C() ρtt 2L2 + ηtt 2L2 + ∇ut 4L2 + ∇ηt 4L2 . √ For any τ ∈ (0, T∗ ), since (t 1/2 ρutt , t 1/2 ηtt ) ∈ L∞ (0, T∗ ; L2 ) by (3.2), there exists some t0 ∈ (τ/2, τ ) such that 

  1 1/2 √ ρ|utt |2 + |ηtt |2 dx(t0 ) ≤ ρutt 2L∞ (0,T ;L2 ) + t 1/2 ηtt 2L∞ (0,T ;L2 ) ≤ C(τ ).

t ∗ ∗ t0 (5.48) Inserting the estimates of VI1 , . . . , VI7 into (5.47), choosing  > 0 small enough and using Lemmas 5.1–5.3, we deduce that 

sup

t0 ≤t≤T

ρ|utt | + |ηtt | 2

2



dx +

T 

 |∇utt |2 + |∇ηtt |2 dxdt ≤ C(τ ).

(5.49)

t0

Using (5.24), (5.33), (5.27), (5.36) and (5.49), we get

sup

τ ≤t≤T

 T 

 |∇utt |2 + |∇ηtt |2 dxdt ≤ C(τ ), (5.50)

∇ut 2H 1 + ∇ηt 2H 1 + ∇ 4 u 2L2 +

due to t0 < τ .

τ

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Finally, we easily deduce from (1.3), (5.24) and (5.50) that sup ∇ 2 η H 2 ≤ C.

(5.51)

τ ≤t≤T

Therefore, combining (5.49)–(5.51) immediately leads to (5.44).

2

6. Proof of Theorems 2.1–2.2 With all the a priori estimates at hand, we are now ready to prove Theorems 2.1–2.2. Proof of Theorem 2.1. From Lemma 3.1, the Cauchy problem (1.1)–(1.7) has a unique classical solution (ρ, u, η) on R3 ×(0, T∗ ] for a small time T∗ > 0. Thus, we shall extend the local classical solution (ρ, u, η) to all time by the a priori estimates, Proposition 4.1 and Lemmas 5.1–5.5. First, by the definition of (4.1)–(4.5), it is easy to get that 

A1 (0) ≤ ε 2 , A5 (0) ≤ M,

A2 (0) ≤ ε, A3 (0) + A4 (0) = 0, and ρ0 ≤ ρ, ˜

due to C0 ≤ ε. Hence, there exists a T1 ∈ (0, T∗ ] such that (4.6) holds for T = T1 . To be continued, we set T ∗ = sup {T | (4.6) holds } ,

(6.1)

then T ∗ ≥ T1 > 0. For any 0 < τ < T ≤ T ∗ , it follows from (1.1), (5.14), (5.15), (5.24), (5.44) and Lemma 3.2 that T 

T 

 

2

∂t ρ|ut | dxdt +

∂t ρ|u∇u|2 dxdt τ

τ

≤C

T 

 |ρt ||ut |2 + ρ|ut ||utt | dxdt

τ

T 

 +C |ρt ||u|2 |∇u|2 + ρ|u||ut ||∇u|2 + ρ|u|2 |∇u||∇ut | dxdt τ

T



∇u L∞ ρ 1/2 ut 2L2 + u L6 ∇ρ L2 ut 2L6 + ρ 1/2 ut L2 ρ 1/2 utt L2 dt ≤C τ

+C

T



∇u 5H 2 + ∇u 5H 2 ∇ρ L2 + ∇u 3H 2 ∇ut L2 + ρ 1/2 ut L2 ∇u 3H 2 dt

τ

≤ C(τ, T ),

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which, together with (5.14) and (5.15), yields ρ 1/2 ut ,

ρ 1/2 u · ∇u ∈ C([τ, T ]; L2 )

and consequently, ρ 1/2 u˙ ∈ C([τ, T ]; L2 ).

(6.2)

T ∗ = ∞.

(6.3)

Next, we claim that

Otherwise, T ∗ < ∞. Then it follows from Proposition 4.1 that (4.7) holds for T = T ∗ . And, we use Lemmas 5.1–5.5 and (6.2) to give that (ρ, u, η)(x, T ∗ ) satisfies (2.1) and (2.2), where g(x) = ρ 1/2 u(x, ˙ T ∗ ). Thus, Lemma 3.1 implies that there exists some T ∗∗ > T ∗ , such that (4.6) holds for T = T ∗∗ , which contradicts (6.1). Hence, (6.3) holds. It directly get from Lemmas 3.1, 5.1–5.5 that (ρ, u, η) is in fact a unique classical solution on R3 × (0, T ] for any 0 < T < T ∗ = ∞. It only needs to prove the large-time behavior (2.6). This can be done as the ones in [13], we only sketch the proof of ∇u(·, t) Lr¯ and ∇η(·, t) Lr¯ here. Setting M(t) 

μ λ

∇u 2L2 + ∇ · u 2L2 . 2 2

Then, multiplying (4.37) by u˙ in L2 and integrating by parts over R3 , we obtain  − (μu + λ∇(∇ · u)) · udx ˙ 

 ˙ − (η + β(ρ − ρ∗ )∇ · u˙ − ρ|u| ˙ 2 dx. = (pF − pF (ρ∗ ) + η)(∇ · u)

(6.4)

Integrating by parts gets that  

 μ d 2

∇u L2 + μ ∂k uj ∂k ui ∂i uj dx − μu · udx ˙ = 2 dt    1 μ d ∂k uj ∂k ui ∂i uj − |∇u|2 (∇ · u) dx

∇u 2L2 + μ = 2 dt 2 

 μ

∇u 2L2 + +C |∇u|3 dx, ≤− t 2    λ

− λ∇(∇ · u) · udx ˙ ≤ − ∇ · u 2L2 + C |∇u|3 dx, t 2 which, inserted into (6.4), yields

   |M  (t)| ≤C ∇u 3L3 + ρ 1/2 u

˙ L2 ˙ 2L2 + pF − pF (ρ∗ ) L2 + η L2 + ρ − ρ∗ L2 ∇ u

 ≤C ∇u L2 ∇u 2L4 + ρ 1/2 u

˙ 2L2 + ∇ u

˙ L2 .

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By (4.65) and (4.76), we get ∞



|M (t)| dt ≤C 2

1

∞

 ˙ 4L2 + ∇u 4L4 + ∇ u

˙ 2L2 dt

ρ 1/2 u

1

 ∞

 ≤C sup ρ

1/2

t≥1

u

˙ 2L2

ρ 1/2 u

˙ 2L2 dt ≤ C, 1

which, together with ∞

∞ |M(t)| dt ≤ C

∇u 2L2 dt ≤ C,

2

1

1

implies

∇u(t) L2 → 0

as t → ∞.

(6.5)

Moreover, by (4.17) and (4.76), we infer from (4.27) that sup ∇u(t) L6 ≤ C, t≥1

which, together with (6.5) and the interpolation inequality, gives that

∇u(t) Lr¯ → 0 as t → ∞,

∀ r¯ ∈ [2, 6).

Finally, multiplying (1.3) by ηt in L2 and integrating by parts over R3 , we obtain |

 d

∇η 2L2 | ≤C ηt 2L2 + ( ∇η 2L2 + ∇ 2 η 2L2 ) ∇u 2L2 + ∇η 2L2 . dt

We infer from (4.18) and (4.76) that ∞

d | ∇η 2L2 |2 dt ≤C dt

1

∞



ηt 4L2 + ∇ 2 η 4L2 + ∇η 4L2 dt 1

 ∞

 ≤C

sup( ηt 2L2 t≥1

+ ∇

2

( ηt 2L2 + ∇ 2 η 2L2 )dt + C

η 2L2 ) 1

≤C, which, together with ∞

∞

∇η 4L2 dt

1

≤C

∇η 2L2 dt ≤ C, 1

(6.6)

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

8705

gives

∇η(t) L2 → 0

as t → ∞.

(6.7)

Thus, we deduce from (6.7), (4.7) and the interpolation inequality that

∇η(t) Lr¯ → 0 as t → ∞,

∀ r¯ ∈ [2, 6).

(6.8)

2

Hence, the proof of Theorem 2.1 is completed.

Proof of Theorem 2.2. By direct calculation, it follows from (4.11)–(4.12) and the Hölder inequality that t

∇η L2 ≤ ∇(L(t)η0 ) L2 +

∇ (L(t − τ )f (τ )) L2 dτ 0

− 32 ( p1 − 12 )− 12

≤C(1 + t)

t

η0 Lp + C

∇ 2 K(t − τ ) ∗ (η(u − ∇)) L2 dτ 0

− 32 ( p1 − 12 )− 12

≤C(1 + t)

t

η0 Lp + C

7

(1 + t − τ )− 4 η(u − ∇) L1 dτ

0 − 32 ( p1 − 12 )− 12

≤C(1 + t)

t

η0 Lp + C

7

(1 + t − τ )− 4 η L2 ( u L2 + ∇ L2 )dτ

0 − 32 ( p1 − 12 )

¯ + t) ≤C(1

,

which, together with (4.9), the interpolation and Sobolev embedding, shows that − 32 ( p1 − 12 )

where θ =

6−q 2q .

¯ + t) ≤ C(1

η Lq ≤ η θL2 η 1−θ L6

f or

Hence, we finish the proof of Theorem 2.2.

2

q ∈ [2, 6],

7. Proof of Theorem 2.3 Setting η˜ = η − η , we can rewrite the system (1.1)–(1.7) and (1.12) as follows: ⎧ ⎪ ρt + ∇ · (ρu) = 0, ⎪ ⎪ ⎪ ⎨ρut + ρu · ∇u + ∇(pF + η) ˜ − μu − λ∇(∇ · u) = −(η˜ + βρ)∇, ∞ u) = −∇ · (η(u −  η ˜ + ∇ · (η ˜ − ∇)) − ∇ · ((η − η∞ )u), η ˜ ⎪ t ⎪

⎪ ⎪

⎩(ρ, u, η) = (ρ0 , u0 , η0 − η ) → (ρ ∞ , 0, 0) as |x| → ∞, ˜ t=0

for some constant vector (ρ ∞ , 0, η∞ ) satisfying ρ ∞ > 0, η∞ > 0.

(7.1)

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Let (ρ, u, η) ˜ be the smooth solution to (7.1) on R3 × (0, T ] with some positive T > 0. In order to derive the time-independent (weighted) necessary estimates and the uniform upper bound of density, we define A˜ 1 (T )  sup η

˜ 3L3 ,

(7.2)

 T 

 2 2 ˜ σ ρ|u| ˙ 2 + |η˜ t |2 + |∇ 2 η| ˜ L2 + ˜ 2 dxdt, A2 (T )  sup σ ∇u L2 + ∇ η

(7.3)

t∈[0,T ]

t∈[0,T ]

A˜ 3 (T )  sup



t∈[0,T ]

0

T 



 2 2 2 2 σ ρ|u| σ 3 |∇ u| ˜ dx + ˙ + |η˜ t | + |∇ η| ˙ 2 + |∇ η˜ t |2 dxdt, 3

(7.4)

0

and ˜ 2L2 ), A˜ 4 (T )  sup ( ∇u 2L2 + ∇ η

t∈[0,T ]

(7.5)

where σ (t)  min {1, t}. Thus, we will prove the following proposition. Proposition 7.1. Under the conditions of Theorem 2.3, there exists another positive constant K˜ depending on μ, λ, ρ ∞ , η∞ , a, γ , ρ, ˜ β, inf ,  H 3 , η∞ − η∞ e− H 2 , M˜ 1 and M˜ 2 such x∈R3

that if (ρ, u, η) ˜ is a smooth solution of (7.1) on R3 × (0, T ] satisfying ⎧ ⎪ ˜ A˜ 1 (T ) ≤ 2ν δ , ⎨ sup ρ ≤ 2ρ, R3 ×[0,T ]

⎪ ⎩A˜ (T ) + A˜ (T ) ≤ 2ν 1/2 , A˜ (σ (T )) ≤ 2K, ˜ 2 3 4

(7.6)

where δ ∈ (0, 34 ), then the following estimates hold ⎧ ⎪ ˜ A˜ 1 (T ) ≤ ν δ , ⎨ sup ρ ≤ 74 ρ, R3 ×[0,T ]

⎪ ⎩A˜ (T ) + A˜ (T ) ≤ ν 1/2 , A˜ (σ (T )) ≤ K, ˜ 2 3 4

(7.7)

provided ν ≤ ν˜ . Throughout this section, for simplicity we denote by C˜ a generic positive constant depending on μ, λ, η∞ , ρ ∞ , a, γ , β, ρ, ˜ inf ,  H 3 , η∞ − η∞ e− H 2 , M˜ 1 and M˜ 2 , but not depending x∈R3

on the time T > 0. To begin with, we will establish the basic energy inequality. Lemma 7.1. Under the conditions of Proposition 7.1, it holds that  

 T ρ|u|2 |η| ˜2 ˜ ˜ 2L2 )dt ≤ Cν, + G(ρ) + dx + ( ∇u 2L2 + ∇ η

2 2 0

(7.8)

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and T

˜ ( ∇u 4L2 + ∇ η

˜ 4L2 )dt ≤ Cν,

(7.9)

0

provided ν ≤ ν˜ 1 . Proof. Using (7.1)2 and (1.12), we get ρut + ρu · ∇u + ρ(

∇pF ∇pF (ρ ) ) + ∇ η˜ − μu − λ∇(∇ · u) = −η∇. ˜ − ρ ρ

(7.10)

Multiplying (7.10) and (7.1)3 by u and η, ˜ respectively, integrating by parts, using (4.22), we obtain d dt

  G(ρ) +

   λ ρ|u|2 μ ˜ · udx, dx + ∇ η˜ · udx + ∇u 2L2 + ∇ · u 2L2 = − η∇ 2 2 2 (7.11) 

1 d ˜ 2L2 − η∞ ∇ η˜ · udx

η

˜ 2L2 + ∇ η

2 dt    = ηu ˜ · ∇ ηdx ˜ − η∇ ˜ · ∇ ηdx ˜ + (η − η∞ )u · ∇ ηdx. ˜

(7.12)

Multiplying (7.11) by η∞ , adding it with (7.12), applying the Hölder inequality, the Cauchy inequality and (2.7), we have   μ λ 1 ρ|u|2 2 ˜ 2L2 η dx + η

˜ L2 + η∞ ( ∇u 2L2 + ∇ · u 2L2 ) + ∇ η

G(ρ) + 2 2 2 2     ∞ η∇ ˜ · udx + ηu ˜ · ∇ ηdx ˜ − η∇ ˜ · ∇ ηdx ˜ + (η − η∞ )u · ∇ ηdx =−η ˜ d dt



∞

 

˜ η

≤C

˜ L6 u L6 ∇

3

L2

+ u L6 η

˜ L3 ∇ η

˜ L2 + η

˜ L6 ∇ L3 ∇ η

˜ L2

+ η − η∞ L3 u L6 ∇ η

˜ L2 ˜ η

≤C ∇ ˜ L2 ∇u L2 ∇

3

L2

+ ∇u L2 η

˜ L3 ∇ η

˜ L2 + ∇ L3 ∇ η

˜ 2L2

+ η − η∞ L3 ∇u L2 ∇ η

˜ L2   η∞ μ ≤ ˜ 2L3 + ∇ L3 + η − η∞ 2L3 ∇ η

∇u 2L2 + C˜ ∇ 2 3 + η

˜ 2L2 , 12 L2 2δ η∞ μ ˜ 2L2 ,

∇u 2L2 + C˜1 ν 3 ∇ η

12   3 choosing ν < ν˜ 1  min 1, (2C˜1 )− 2δ and using (1.5), we obtain

≤

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S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

d dt



η∞

  G(ρ) +

  μ 1 1 ρ|u|2 dx + η

˜ 2L2 + η∞ ∇u 2L2 + ∇ η

˜ 2L2 ≤ 0, 2 2 12 2

(7.13)

integrating it over (0, T ), we have  

 T ˜2 ρ|u|2 |η| ˜ ˜ 2L2 )dt ≤ Cν. + dx + ( ∇u 2L2 + ∇ η

G(ρ) + 2 2

(7.14)

0

As the ones in (4.19), by (7.6), we also prove (7.9). The next lemma is concerned with the estimate of A˜ 1(T ). Lemma 7.2. Under the conditions of Proposition 7.1, it holds that A˜ 1 (T ) +

T

η

˜ 3L9 dt ≤ ν δ ,

(7.15)

0

provided ν ≤ ν˜ 2 . Proof. Multiplying (7.1)3 by 3η˜ 2 , integrating by parts over R3 , using the Hölder inequality and the Cauchy inequality, we obtain  d 3 ˜ η| ˜ 2 dx

η

˜ L3 + 6 η|∇ dt     ˜ L6 η

˜ L3 + η˜ 2 |u||∇ η|dx ˜ + η˜ 2 |∇||∇ η|dx ˜ ≤C˜ ∇u L2 η

+ C˜



η|(η ˜  − η∞ )||u||∇ η|dx ˜

3/2 2 ˜ ˜ 2 ≤C ∇u

+ C ∇ η

˜ 2L2 η

˜ 2L3 + C u

˜ L2 η

˜ L9/2 L6 η˜ |∇ η|

L2 1

˜ ˜ 2 + C ∇

˜ L3 η

˜ L6 ∇ η

˜ L2 + C u

˜ L2 η

˜ L3 η − η∞ L6 L∞ η

L6 η˜ |∇ η|

1

1/2

2 2 ˜ ˜ η

˜ ˜ η

≤C ∇u

+ C ∇ ˜ 2L2 η

˜ 2L3 + C ∇u

η

˜ 3L9/2 + η˜ 2 |∇ η|

˜ 2L2 + C ∇ ˜ 2L2 L2 L2 1

2 ˜ + C ∇u

η

˜ L3 η − η∞ 2L6 . L2

(7.16)

To deal with the right-hand side of (7.16), we use Lemma 3.2 to get that ⎧ ⎨ η

˜ ⎩

L9/2

η

˜ 3L9

˜ η

≤ C

˜ L3 η

˜ L9 , 1/2

= η˜ 3/2 2L6

1/2

1

1

η

˜ L3 ≤ η

˜ L2 2 ∇ η

˜ L2 2 ,

˜ ˜ η˜ 1/2 ∇ η

≤ C ∇( η˜ 3/2 ) 2L2 ≤ C

˜ 2L2 .

(7.17)

Thus, putting (7.17) into (7.16), using (2.7), (7.6) and the Cauchy–Schwarz inequality, we find d 2 ˜ ˜ 3L9 ≤ C( ∇u

+ ∇u 4L2 + ∇ η

˜ 2L2 ),

η

˜ 3L3 + η

L2 dt

(7.18)

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integrating it over (0, T ), using (7.8)–(7.9), we get T sup 0≤t≤T

η

˜ 3L3

+

3/2 ˜ η˜ 0 3 3 + Cν ˜ ≤ C

˜ η˜ 0 3/2 ˜ ≤ C˜ 2 ν 3/4 .

η

˜ 3L9 dt ≤ C

∇ η˜ 0 L2 + Cν L L2

(7.19)

0



−

Therefore, choosing ν ≤ ν˜ 2  min ν˜ 1 , C˜ 2

1 3 −δ 4

 , we immediately obtain (7.15) from (7.19).

Next, we denote ˜ F˜  ρ−1 [(μ + λ)divu − (pF (ρ) − pF (ρ )) − η].

(7.20)

It follows from (1.12) that ρ−1 (∇pF + (η˜ + βρ)∇) =ρ−1 [∇(pF − pF (ρ ) + η) ˜ − (βρ )−1 (η˜ + β(ρ − ρ ))∇pF (ρ )] =∇[ρ−1 (pF − pF (ρ ) + η)] ˜ + ρ−2 [pF − pF (ρ ) + η˜ − β −1 (η˜ + β(ρ − ρ ))pF (ρ )]∇ρ , (7.21) which, together with (7.1)2 and (1.12), gives that ρ−1 ρ u˙ − ∇ F˜ + μcurl(ρ−1 curlu) = − [(λ + μ)(∇ · u)∇ρ−1 − μ∇ρ−1 × (curlu)] + [pF − pF (ρ ) + η˜ − β −1 (η˜ + β(ρ − ρ ))pF (ρ )]∇ρ−1 ˜1 +G ˜ 2, G

(7.22)

which leads to ˜1 −G ˜ 2 ),

F˜ = ∇ · (ρ−1 ρ u˙ − G

(7.23)

and μ (ρ−1 curlu) =μ∇(∇ · (ρ−1 curlu)) − μcurlcurl(ρ−1 curlu) ˜1−G ˜ 2 ). =μ∇(curlu · ∇ρ−1 ) + curl(ρ−1 ρ u˙ − G

(7.24)

Therefore, applying the same proof as the one in Lemma 4.3, we also arrive at Lemma 7.3. Under the conditions of Proposition 7.1, furthermore assume that F˜ = F˜ (x, t) is the modified effective viscous flux defined in (7.20). Then it holds that ˜

∇ F˜ L2 + ∇(ρ−1 curlu) L2 ≤ C( ρ u

˙ L2 + ∇u L3 + η

˜ L3 + ρ − ρ 2L6 ),



∇u L6 ≤ C˜ ρ u

˙ L2 + ∇u L2 + ρ − ρ L6 + ρ − ρ 2L6 + η

˜ L2 + ∇ η

˜ L2 .

(7.25) (7.26)

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Lemma 7.4. ˜ + C˜ A˜ 2 (T ) ≤ Cν

t σ ∇u 3L3 ds,

(7.27)

0

˜ + CA ˜ 3 (T ) + C˜ A˜ 3 (T ) ≤ Cν

t σ 3 ∇u 4L4 ds,

(7.28)

0

provided ν ≤ ν˜ 3 . Proof. Multiplying the equation (7.1)3 by σ m (η˜ t + η) ˜ and integrating the result equation with respect to x over R3 , we deduce that d m ˜ 2L2 ) + σ m ( η˜ t 2L2 + ∇ 2 η

˜ 2L2 ) (σ ∇ η

dt  ˜ ≤C σ m |∇ · (η(u ˜ − ∇))|2 dx + mσ m−1 σ  ∇ η

˜ 2L2   + C˜ σ m |∇u|2 dx + C˜ σ m |∇ · ((η − η∞ )u)|2 dx ˜ m ( η

≤Cσ ˜ 2L∞ ∇u 2L2 + u 2L6 ∇ η

˜ 2L3 + ∇ η

˜ 2L2 ∇ 2H 2 + η

˜ 2L∞  2L2 ) ˜ m ∇u 2 2 + mσ m−1 σ  ∇ η

˜ 2L2 + Cσ L ˜ m ∇(η − η∞ ) 2 3 u 2 6 + Cσ ˜ m (η − η∞ ) 2L∞ ∇u 2 2 + Cσ L L L 1 ˜ m ( ∇ η

≤ σ m ∇ 2 η

˜ 2L2 + Cσ ˜ 2L2 ∇u 4L2 + ∇ η

˜ 2L2 + ∇u 2L2 ) + mσ m−1 σ  ∇ η

˜ 2L2 , 2 that is d m ˜ 2L2 ) + σ m ( η˜ t 2L2 + ∇ 2 η

˜ 2L2 ) (σ ∇ η

dt m−1  ˜ ˜ m ( ∇ η

˜ 2 2 ∇u 4 2 + ∇u 2 2 ) + C(mσ σ + σ m ) ∇ η

˜ 2 2. ≤Cσ L

L

L

L

(7.29)

Choosing m = 1 in (7.29), integrating it over [0, T ], using Gronwall’s inequality and Lemma 7.1, we find that t sup 0≤t≤T

2 (σ ∇ η(t)

˜ )+ L2

2 ˜ σ ( η˜ t (s) 2L2 + ∇ 2 η(s)

˜ )ds ≤ Cν. L2

(7.30)

0

It follows from (7.1)2 and (1.12) that ρ u˙ + ∇(pF − pF (ρ ) + η) ˜ − μu − λ∇(∇ · u) = −(η˜ + β(ρ − ρ ))∇, which is the same as (4.37). So we also deduce from (7.31) that

(7.31)

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717 1 μ λ ( σ m ∇u 2L2 + σ m ∇ · u 2L2 )t + σ m ρ 2 u

˙ 2L2 2 2 d 2 ˜ ˜ 2 σ 2m−1 σ  ν + Cσ ˜ m η˜ t 2 2 ≤ I˜0 + C( ∇u

+ η

˜ 2L2 ) + Cm L2 L dt m 6 m 3 ˜ ∇ η

˜ ∇u 3 , + Cσ ˜ 2 + Cσ

L

L

8711

(7.32)

where |I˜0 |  | ≤

 σ m ((∇ · u)(pF − pF (ρ )) − (β(ρ − ρ ) + η)u ˜ · ∇ + (∇ · u)η) ˜ dx|

μ m ˜ σ ∇u 2L2 + Cν, 12

and 1

(σ m ρ 2 u

˙ 2L2 )t + σ m ∇ u

˙ 2L2 ˜ m ( ∇u 2 2 + ∇ η

≤Cσ ˜ 2L2 + η˜ t 2L2 + ∇ η

˜ 4L2 + ∇u 4L2 + ∇ η

˜ 2L2 ∇ 2 η

˜ 2L2 ) L m−1  ˜ ˜ m ∇u 4 4 . + C(mσ σ + σ m ) ρ 2 u

˙ 2L2 + Cσ L 1

(7.33)

Differentiating (7.1)3 with respect to t, we have ˜ t − η˜ t ∇) − η˜ t = −η∞ ∇ · ut − ∇ · ((η − η∞ )ut ), η˜ tt + ∇ · (η˜ t u + ηu

(7.34)

multiplying it by σ m η˜ t , integrating over R3 and using the fact ut = u˙ − u · ∇u, we get m 1 d m (σ η˜ t 2L2 ) + σ m ∇ η˜ t 2L2 − σ m−1 σ  η˜ t 2L2 2 dt 2   m = − σ (η˜ t (∇ · u) + ∇ η˜ t u) η˜ t dx − σ m (∇ η˜ · u˙ + η(∇ ˜ · u)) ˙ ηt dx   ˜ · (u · ∇u)))η˜ t dx − σ m η˜ t ∇ · ∇ η˜ t dx + σ m (∇ η˜ · (u · ∇u) + η(∇   ˙ + η∞ σ m η˜ t (∇ · (u · ∇u))dx − η∞ σ m η˜ t (∇ · u)dx  + σ m (η − η∞ )(u˙ − u · ∇u) · ∇ η˜ t dx 

7 

 IIi .

(7.35)

i=1

Similar to the procedure of (4.56)–(4.59), one obtains that σm ˜ m η˜ t 2 2 ∇u 4 2 ,

∇ η˜ t 2L2 + Cσ L L 24 m m σ σ ˜ m ∇ η

 ˜ 4L2 η˜ t 2L2 ,

∇ u

˙ 2L2 +

∇ η˜ t 2L2 + Cσ II2 ≤ 12 24  II1 ≤

(7.36) (7.37)

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S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717



σm ˜ m ∇ η

˜ 2L2 ∇u 2L2 ρ u

˜ 2L2 + 1 , (7.38)

∇ η˜ t 2L2 + Cσ ˙ 2L2 + ∇u 2L2 + ∇ η

24

 II3 ≤ and

 II4 ≤

σm ˜ m η˜ t 2 2 .

∇ η˜ t 2L2 + Cσ L 24

(7.39)

For other terms, we also obtain that σm σm ˜ m η˜ t 2 2 + Cσ ˜ m u 2 6 ∇u 2 3

∇ u

˙ 2L2 +

∇ η˜ t 2L2 + Cσ L L L 12 24 m m 2 4 σ σ 3) 3 ˜ m η˜ t 2 2 + Cσ m ∇u (2+ ≤

∇u

∇ u

˙ 2L2 +

∇ η˜ t 2L2 + Cσ 2 L L L4 12 24 σm σm ˜ m η˜ t 2 2 + Cσ ˜ m ∇u 4 2 ≤

∇ u

˙ 2L2 +

∇ η˜ t 2L2 + Cσ L L 12 24 ˜ m ∇u 4 4 , + Cσ (7.40) L

 II6 ≤ II5 + 

˜ m η − η∞ L3 u

˜ m η − η∞ L6 u L6 ∇u L6 ∇ η˜ t L2  ˙ L6 ∇ η˜ t L2 + Cσ II7 ≤Cσ ≤

σm σm

∇ u

˙ 2L2 + C˜ 3 σ m η − η∞ 2L3 ∇ η˜ t 2L2 +

∇ η˜ t 2L2 12 24 

2 2 2 ˜ m η − η∞ 2 6 ∇u 2 2 ρ u

+ Cσ +

∇u

+

∇ η

˜ + 1 . ˙ 2 2 2 L L L L L

 Putting (7.36)–(7.41) into (7.35), choosing ν ≤ ν¯ 3  min ν¯ 2 ,

1 4C˜ 3

(7.41)

 , we get

(σ m η˜ t 2L2 )t + σ m ∇ η˜ t 2L2

 ˜ m η˜ t 2 2 ( ∇u 4 2 + ∇ η

≤C˜ mσ m−1 σ  + σ m η˜ t 2L2 + Cσ ˜ 4L2 ) L L ˜ m ∇u 4 2 + Cσ ˜ m ∇u 4 4 + Cσ L L

 ˜ m ( ∇ η

+ Cσ ˜ 2L2 ∇u 2L2 + ∇u 2L2 ) ρ u

˙ 2L2 + ∇u 2L2 + ∇ η

˜ 2L2 + 1 +

σm

∇ u

˙ 2L2 . 2

(7.42)

Hence, similar to the proof of (4.62) and (4.64), we also obtain (7.27) and (7.28) from (7.30), (7.32), (7.33) and (7.42). 2 We use (7.20) and (7.1)1 to get that (ρ − ρ )t +

ρ ρ 2 F˜ ρ η˜ (pF (ρ) − pF (ρ )) = −∇ · (u(ρ − ρ )) − u · ∇ρ −  − , μ+λ μ+λ μ+λ

(7.43)

which is the same as (4.66). Therefore, applying the same procedure of Lemma 4.5, we have

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

8713

Lemma 7.5. Under the conditions of Proposition 7.1, it holds that T

 ˜ σ 3 ∇u 4L4 + ρ − ρ 4L4 + F˜ 4L4 + η

˜ 4L4 dt ≤ Cν,

(7.44)

0

  −2 . provided ν ≤ ν˜ 4  min ν˜ 3 , (2C˜ 4 ) Lemma 7.6. Under the conditions of Proposition 7.1, then there exists another positive constant K˜ depending on μ, λ, γ , a, ρ, ˜ β, ν,  H 3 , η∞ − η∞ e− H 2 , M˜ 1 and M˜ 2 such that A˜ 4 (σ (T )) +

σ(T )

1  ˜ ˙ 2L2 + η˜ t 2L2 + ∇ 2 η

˜ 2L2 dt ≤ K,

ρ 2 u

(7.45)

0

provided ν ≤ ν˜ 5 . Proof. Taking m = 0 in (7.29) and (7.32), applying the argument of (4.75), yields that σ(T )



A4 (σ (T )) + 2

 1 ˙ 2L2 + η˜ t 2L2 + ∇ 2 η

˜ 2L2 dt

ρ 2 u

0 σ(T ) 1 K˜ 2 ≤ + C˜ 5 ν A˜ 4 (σ (T )) +

ρ 2 u

˙ 2L2 dt, 2

(7.46)

0

  ˜ 5 )−1 , leads to (7.45). choosing ν ≤ ν˜ 5  min ν˜ 4 , (8KC

2

Next, we close the estimates of A˜ 2 (T ) and A˜ 3 (T ). Lemma 7.7. Under the conditions of Proposition 7.1, it holds that A˜ 2 (T ) + A˜ 3 (T ) ≤ ν 1/2 ,

(7.47)

provided ν ≤ ν¯ 6 . Proof. As to the proof of (4.76). Putting (7.27) and (7.28) together, using (7.44), we show that ˜ + C˜ A˜ 2 (T ) + A˜ 3 (T ) ≤ Cν

σ(T )

0



 36

− provided ν ≤ ν˜ 6 = min ν˜ 5 , C˜ 6 7 .

2

3

3

σ ∇u L2 2 ∇u L2 6 dt ≤ C˜ 6 ν 36 ≤ ν 2 , 25

1

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S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

Lemma 7.8. Under the conditions of Proposition 7.1, it holds that T sup

t∈[0,T ]

( ∇u 2L2

+ ∇ η

˜ 2L2 ) +

˜ ( ρ 2 u

˙ 2L2 + η˜ t 2L2 + ∇ 2 η

˜ 2L2 )dt ≤ C, 1

(7.48)

0

and T sup

t∈[0,T ]

σ ρ u

˙ 2L2

+

˜ σ ∇ u

˙ 2L2 dt ≤ C,

(7.49)

0

provided ν ≤ ν˜ 6 . Proof. Adding (7.45) with (7.47), choosing m = 1 in (7.33), using the same processes of (4.79) and (4.80), respectively, we can obtain (7.48) and (7.49). Here, we omit the details. Finally, we can prove the uniform upper bound of the density ρ. Lemma 7.9. Under the conditions of Proposition 7.1, it holds that 7 sup ρ L∞ ≤ ρ, ˜ 4 0≤t≤T 

provided ν ≤ ν˜  min ν˜ 8 ,

ρ˜ 4C˜

2 

(7.50)

.

Proof. Similar to the proof of Lemma 4.9, for completeness we sketch it here. We use (7.1)1 and (7.1)2 to get that Dt ρ = g(ρ) + b (t) where aρ γ g(ρ)  − (ρ γ − ρ ), μ+λ

1 b(t)  − μ+λ

t

ρ(F˜ + η)ds, ˜

0

and ˜ F˜  (μ + λ)(∇ · u) − (pF − pF (ρ )) − η.

(7.51)

One obtains lim g(ρ) = −∞ by the definition of g. It follows from (7.1)2 and (1.12) that ρ→∞

F˜ = ∇ · (ρ u˙ + (η˜ + β(ρ − ρ ))∇) . Applying Lp -estimate for (7.52), we have

(7.52)

S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717 1

8715

3

˜ u

˜ η

˜ − ρ L12 ∇ L6

∇ F˜ L4 ≤C ρ ˙ L4 2 ρ u

˙ L4 6 + C

˜ L4 ∇ L∞ + C ρ 3

˜ − 8 ∇ u

˜ 12 + C ∇ ˜ η

≤Cσ ˙ L4 2 + Cν ˜ 2L2 , 1

1

1

6

(7.53) 9

12

˜ F˜ 7 2 ∇ F˜ 7 4 ≤ Cσ ˜ − 28 ∇ u

˜ 14 + C ∇ ˜ η

F˜ L∞ ≤ C

˙ L142 + Cν ˜ L72 . L L 3

1

(7.54)

For 0 ≤ t1 < t2 ≤ σ (T ) ≤ 1, we infer from (7.54), (7.47) and (7.49) that |b(t2 ) − b(t1 )| ≤ C˜

σ(T )

1 1 1 ˜ 14 + C˜ A˜ 3 (T ) 14 ≤ Cν ˜ 28 . ( F˜ L∞ + η

˜ L∞ )dt ≤ Cν

(7.55)

0

Thus, for t ∈ [0, σ (T )], N0 , N1 and ζ in Lemma 3.3 can be chosen as follows: ˜ 28 , N0 = Cν 1

N1 = 0,

ζ = ρ. ¯

Since it holds that 0 < ρ ≤ ρ ≤ ρ¯ < ∞ in (1.17) and g(ξ ) = −

 aξ  γ ξ − ρ¯ γ ≤ −N1 = 0, μ+λ

∀ ξ ≥ ζ = ρ, ¯

therefore, it follows from (3.6) that 1 ˜ 28 ≤ 3 ρ,

ρ L∞ ≤ max{ρ, ¯ ρ} ˜ + N0 ≤ ρ˜ + Cν ˜ 2 0≤t≤σ (T )

sup

(7.56)

where 



ρ˜ ν ≤ ν˜ 7  min ν˜ 6 , 2C˜

28  .

On the other hand, for σ (T ) ≤ t1 < t2 ≤ T , we use (1.19), (7.8), (7.48), (7.53)1 , (7.47) and (7.49) to get that |b(t2 ) − b(t1 )| ≤C˜

t2

( F˜ L∞ + η

˜ L∞ )dt

t1

≤C˜

T

 1 1 ˜ 14 + σ 3 ( ρ 2 u

˙ 2L2 + ∇ u

˙ 2L2 )dt + Cν

σ (T )

˜ 2+ ≤Cν 1

 1 a ˜ 2 (t2 − t1 ) + Cν 2(μ + λ)

a (t2 − t1 ), μ+λ



where ν in the last inequality of (7.57) satisfying ν ≤ ν˜ 8  min ν˜ 7 , t ∈ [σ (T ), T ], N0 , N1 and ζ in Lemma 3.3 can be chosen as follows:

(7.57) a ˜ 2C(μ+λ)

14 

. Thus, for

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S. Ding et al. / J. Differential Equations 263 (2017) 8666–8717

˜ 1/2 , N0 = Cν

N1 =

a , μ+λ

ζ = ρ¯ + 1.

Since it holds that g(ξ ) = −

 aξ  γ a ξ − ρ¯ γ ≤ −N1 = − , μ+λ μ+λ

∀ ξ ≥ ζ = ρ¯ + 1,

then it follows from Lemma 3.3 and (7.56) that 

sup σ (T )≤t≤T

ρ

L∞

 3ρ˜ 3ρ˜ ˜ 1/2 ≤ 7 ρ, ≤ max , ρ¯ + 1 + N0 ≤ + Cν ¯ 2 2 4

(7.58)

where 



ρ˜ ν ≤ ν˜  min ν˜ 8 , 4C˜

2  .

Therefore, the combination of (7.56) and (7.58) yields (7.50).

2

Proof of Proposition 7.1. By virtue of Lemmas 7.1, 7.6, 7.8–7.9, the proof of Proposition 7.1 is complete. 2 Next, the time-dependent higher norm estimates of the smooth solution (ρ, u, η) ˜ can be established in similar arguments as used in section 5. Furthermore, following the methods in [11], ˜ satwe can show that the Cauchy problem (7.1) has a unique local classical solution (ρ, u, η) isfying the same regularity in (3.2). Finally, applying the same processes as that in the proof of Theorem 2.1, Theorem 2.3 can be obtained. We omit the details here for simplicity. 2 Acknowledgments Ding was supported by the National Natural Science Foundation of China (Grant No. 11371152, 11771155 and 11571117), and by the Guangdong Province Natural Science Foundation (Grant No. 2017A030313003). Huang was supported by the Foundation for Distinguished Young Talents in Higher Education of Guangdong (Grant No. 2015KQNCX095). Wen was supported by the National Natural Science Foundation of China (Grant No. 11671150, 11722104), and by GDUPS 2016 and the Fundamental Research Funds for the Central Universities of China (Grant No. D2172260). References [1] J. Ballew, Low Mach number limits to the Navier Stokes Smoluchowski system, in: Hyperbolic Problems: Theory, Numerics, Applications, in: AIMS Series on Applied Mathematics, vol. 8, 2014, pp. 301–308. [2] J. Ballew, Mathematical Topics in Fluid-Particle Interaction, PHD dissertation, University of Maryland, USA, 2014. [3] J. Ballew, K. Trivisa, Weakly dissipative solutions and weak-strong uniqueness for the Navier Stokes Smoluchowski system, Nonlinear Anal. 91 (2013) 1–19. [4] J.T. Beale, T. Kato, A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Comm. Math. Phys. 94 (1984) 61–66.

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