Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3

Leilei Tong

doi:10.1515/anona-2023-0131

Open Access Published by De Gruyter February 24, 2024

Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ³

Leilei Tong

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2023-0131

Abstract

The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution ( ρ s ( x ) , 0 , 0 ) T is in a small neighborhood of the constant state ( ρ ¯ , 0 , 0 ) T at infinity, we also obtain the time decay rates of the solution by the combination of the energy method and the linear L p - L q decay estimates.

Keywords: Navier-Stokes-Smoluchowski equations; global classical solution; decay estimates; energy method

MSC 2010: 35Q35; 76N10; 35B40; 46E35

1 Introduction

The compressible Navier-Stokes-Smoluchowski system can be comprehended as a fluid–particles interaction model, which has broad applications in hemodynamics [20,38], sprays and aerosols [28,43], and sedimentation [4]. The particles in the system are assumed to be light compared with the fluid and will tend to bubble upward due to buoyancy effects [15]. In the condition of high concentration of particles, the interaction between the particles can be described by a potential. The particles are transported by a fluid and described by a probability distribution that is responsible for the Smoluchowski equation [10]. Mathematically, the evolution of disperse suspensions of the particles in the viscous compressible fluid can be described by the Navier-Stokes equations coupled to the Smoluchowski equation via a drag force, which take the following form [2,7,15]:

(1.1) ρ t + div ( ρ u ) = 0 , ( ρ u ) t + div ( ρ u ⊗ u ) + ∇ ( P + η ) = − ( η + β ρ ) ∇ Φ + μ Δ u + ( λ + μ ) ∇ div u , η t + div ( η u − η ∇ Φ ) − Δ η = 0 .

The fluid density ρ = ρ ( x , t ) , the fluid velocity u = u ( x , t ) , and the particle density η = η ( x , t ) in equations (1.1) are unknown quantities. The external potential Φ = Φ ( x ) reflecting the effects of gravity and buoyancy is time independent. The fluid pressure P = P ( ρ ) is a smooth function and satisfies P ′ ( ρ ) > 0 for ρ > 0 . The positive parameter β is a constant representing the differences in how the external potential Φ ( x ) affects the particles and the fluid. The viscosity coefficients μ and λ satisfy

μ > 0 , 2 μ + 3 λ ⩾ 0 .

In case the particle density η is absent, equation (1.1) will be reduced to the compressible Navier-Stokes equations. There are plenty of studies on the Navier-Stokes equations without external force, for instance, the existence of weak solutions [8,19,22], strong solutions [9,32,46], classical solutions [21,25,26,33,35,40,45], the convergence rates [11,23,31,37,44], as well as the asymptotic stability results [14,49], and the references therein.

For the Navier-Stokes equations with a potential external force in three-dimensional space, Matsumura and Nishida [41] investigated the initial boundary value problem with small external force and initial perturbation on the exterior domain and the half space, and the global classical solution was proved to exist uniquely. Duan et al. [16,17] obtained the optimal convergence rates under the smallness conditions on the initial perturbation of the stationary solution and the potential force in R 3 . Under the assumptions of large external potential forces and discontinuous initial data, Li and Matsumura [34] established the global weak solutions with small initial perturbation of the stationary solution in R 3 . In addition, some other interesting progress has been made for the related model, see [1,39] and the references therein.

We take a brief review on the researches about the Navier-Stokes-Smoluchowski equations which have come into people’s note in recent years. Neglecting the viscosity terms in equation (1.1), this system was derived from a Vlasov-Fokker-Planck equation by formal hydrodynamic limit [5]. Ballew and Trivisa [3] analyzed the local existence of the weakly dissipative solutions to the Navier-Stokes-Smoluchowski equations in 3 D bounded domain and established the weak-strong uniqueness result by the relative entropy method. The studies on the existence of global weak solutions and the large time behavior were carried out by Carrillo et al. [6] in three-dimensional domain, which may be unbounded. The global weak solution in R 2 was proposed by Constantin and Masmoudi [10] via a deteriorating regularity estimate. The local strong solutions to the Cauchy problem of the compressible Navier-Stokes-Smoluchowski equations with vacuum were investigated by Liu [36] in R 2 and by Yang [50] in 2 D bounded domain; furthermore, the global existence of the strong solution with large initial data, which may contain vacuum in R 2 , was obtained in [27]. Fang et al. [18] investigated the global classical large solution with initial density containing vacuum in 1 D space. Huang et al. [24] studied the Cauchy problem of equations (1.1) with vacuum in R 3 and obtained the local classical solutions. Moreover, the global classical solution to the Cauchy problem of equations (1.1) in the presence of vacuum in R 3 was established in [13] when the initial data were of small energy around the steady state ( ρ s ( x ) , 0 , η s ( x ) ) T . Chen et al. [7] obtained the global classical solution for the Cauchy problem of equation (1.1) with the stationary solution ( ρ s ( x ) , 0 , 0 ) T in a small neighborhood of ( ρ ¯ , 0 , 0 ) T and shown the time decay rates of the classical solution approaching to the stationary solution in R 3 . Under further smallness conditions on the external potential, Ding et al. [12] improved the work in [7] to the case that the stationary solution of the particle density was nonconstant.

This article is devoted to considering the Cauchy problem of the Navier-Stokes-Smoluchowski equations (1.1) in R 3 and looking for the global classical solution near the stationary solution ( ρ s ( x ) , 0 , 0 ) T ; moreover, the decay rates of the solution are also investigated on the condition that the stationary solution is in a small neighborhood of ( ρ ¯ , 0 , 0 ) T . We complement equation (1.1) with the initial data

(1.2) ( ρ , u , η ) T ∣ t = 0 = ( ρ 0 , u 0 , η 0 ) T → ( ρ ¯ , 0 , 0 ) T as ∣ x ∣ → + ∞

for the positive constant ρ ¯ . Similar to the statement in [7,41], there exists the steady state solution ( ρ s , 0 , 0 ) T of equation (1.1) with ρ s > 0 such that

(1.3) ∇ P ( ρ s ) = − β ρ s ∇ Φ ,

which implies

(1.4) − β Φ = ∫ ρ ¯ ρ s P ′ ( ξ ) ξ d ξ ,

and the stationary solution ( ρ s , 0 , 0 ) T satisfies

(1.5) ∥ ρ s − ρ ¯ ∥ H 3 ⩽ C ∥ Φ ∥ H 3 .

We define the perturbation by

(1.6) n = ρ − ρ s , u = u − 0 , η = η − 0 ,

then problem (1.1) can be reformulated into the perturbed form of

(1.7) ∂ t n + div ( ρ s u ) = F 1 , ∂ t u + ∇ ( h ′ ( ρ s ) n ) − 1 ρ s ( μ Δ u + ( μ + λ ) ∇ div u ) + ∇ η ρ s = F 2 , η t − Δ η = F 3 , ( n , u , η ) T ∣ t = 0 = ( n 0 , u 0 , η 0 ) T = ( ρ 0 − ρ s , u 0 , η 0 ) T ,

with

F 1 = − div ( n u ) , F 2 = − u ⋅ ∇ u − ∇ ℛ − 1 ρ s − 1 n + ρ s ( μ Δ u + ( μ + λ ) ∇ div u ) + 1 ρ s − 1 n + ρ s ∇ η − η ∇ Φ n + ρ s , F 3 = − div ( η u − η ∇ Φ ) ,

where

h ( z ) = ∫ ρ ¯ z P ′ ( s ) s d s .

By Taylor expansion, we have

h ( n + ρ s ) = h ( ρ s ) + h ′ ( ρ s ) n + ℛ ,

with

(1.8) ℛ = ∫ ρ s n + ρ s h ′ ′ ( s ) ( n + ρ s − s ) d s .

Notation. We use the notation x ≲ y to express x ⩽ C y with some positive constant C > 0 . The norm ∥ ( x , y ) ∥ L p is short for ∥ x ∥ L p + ∥ y ∥ L p .

Our first result in this article is concerned with the global existence of the classical solution near the steady state ( ρ s , 0 , 0 ) T .

Theorem 1.1

Under the condition that ∥ ( n 0 , u 0 , η 0 ) ∥ H k 2 + ∥ ∇ Φ ∥ L 2 2 + ∥ η 0 ∥ L p 2 ⩽ δ with k ⩾ 3 and 1 ⩽ p < 6 ∕ 5 , where the positive constant δ is sufficiently small, there exists a unique global solution ( n , u , η ) T to the system (1.7) such that for all t ⩾ 0 ,

(1.9) ∥ ( n , u , η ) ( t ) ∥ H k 2 + ∫ 0 t ( ∥ ∇ n ( τ ) ∥ H k − 1 2 + ∥ ∇ ( u , η ) ( τ ) ∥ H k 2 ) d τ ⩽ C ∥ ( n 0 , u 0 , η 0 ) ∥ H k 2 + ∥ η 0 ∥ L p 2 .

We want to complete the proof of Theorem 1.1 based on the energy method. Unfortunately, combined with the dissipation estimates of the solution ( n , u , η ) T , we just obtain the energy estimate (3.48), and it cannot be closed due to the appearance of the linear term ∇ η ρ s in the momentum equation (1.7)₂, which will cause trouble in providing small quantity δ in front of η and its higher order derivative, see Lemma 3.1. We note that the third equation in equation (1.7) is a parabolic one. By the spectral analysis, it is able to deduce the decay rate of η in L 2 norm, which is integrable with respect to t , for the reason that we have restricted the value 1 ⩽ p < 6 ∕ 5 . Therefore, the uniform estimate (1.9) of the solution ( n , u , η ) T can be established, which along with the a priori estimates give the global classical solution via a standard continuity argument.

Remark 1.2

In Theorem 1.1, the conditions imposed on the initial perturbation and the external potential are weaker compared with the existence results in [7]. We only assume that the gradient of the external potential ∇ Φ in L 2 norm is sufficiently small; the smallness condition on the term ∥ ( 1 + ∣ x ∣ ) ∇ Φ ∥ L 2 ∩ L 3 is removed.

Under the assumptions of Theorem 1.1, if further assumptions are imposed on Φ and the initial data ( n 0 , u 0 ) , we can obtain the following result about the time convergence rate of the solution ( n , u , η ) T .

Theorem 1.3

Under the same conditions of Theorem 1.1, if further assume that ∥ Φ ∥ H k + 1 + ∥ ∇ Φ ∥ L r with 1 ⩽ r < 6 ∕ 5 is sufficiently small and ∥ ( n 0 , u 0 ) ∥ L p with 1 ⩽ p < 6 ∕ 5 is finite, then for 0 ⩽ ℓ ⩽ 1 , we have

(1.10) ∥ ∇ ℓ ( n , u , η ) ( t ) ∥ H k − ℓ ⩽ C 0 ( 1 + t ) − 3 2 1 max ( p , r ) − 1 2 − ℓ 2 .

Remark 1.4

In Theorem 1.3, owing to the conditions imposed on the external potential Φ and equation (1.5), it implies the smallness of ∥ ρ s − ρ ¯ ∥ H 3 .

It’s difficult to obtain the decay rates of the system (1.7) which is an evolution one with variable coefficient ρ s , for the reason that the Fourier transform fails. In the process of proving Theorem 1.3, we reformulate equation (1.7) in the form of equation (4.1) and establish the energy estimates again as shown in Lemmas 4.2–4.4. By the combination of the linear decay property and the energy estimates, we obtain the decay results (1.10) of the solution ( n , u , η ) T .

The structure of the article is as follows: in Section 2, some analysis tools are prepared, which will be helpful in establishing the energy estimates of the solution ( n , u , η ) T ; the existence of the global solution is proved in Section 3 based on the refined energy estimates shown in Lemmas 3.1–3.4; in Section 4, we reformulate the Navier-Stokes-Smoluchowski equations into the form of equation (4.1) and establish the energy estimates again to complete the proof of Theorem 1.3.

2 Preliminaries

This section is mainly about the lemmas to be used in establishing the energy estimates in Sections 3 and 4. The Gagliardo-Nirenberg’s inequality, which is also called the interpolation inequality, will be stated in the following lemma.

Lemma 2.1

If the constants 0 ⩽ m , α ⩽ l , it holds for 2 ⩽ p ⩽ ∞ ,

(2.1) ∥ ∇ α f ∥ L p ≲ ∥ ∇ m f ∥ L 2 1 − ϑ ∥ ∇ l f ∥ L 2 ϑ ,

with the superscript 0 ⩽ ϑ ⩽ 1 and the constant α satisfies

α + 3 1 2 − 1 p = m ( 1 − ϑ ) + l ϑ .

Note that if p = ∞ , it is required that 0 < ϑ < 1 , m ⩽ α + 1 , and l ⩾ α + 2 .

Proof

The proof of this lemma can be seen in the study by Nirenberg [42].□

Lemma 2.2

The commutator notation is defined as

(2.2) [ ∇ l , g ] h = ∇ l ( g h ) − g ∇ l h

for the integer l ⩾ 1 . Then, the commutator estimates

(2.3) ∥ [ ∇ l , g ] h ∥ L p 0 ≲ ∥ ∇ g ∥ L p 1 ∥ ∇ l − 1 h ∥ L p 2 + ∥ ∇ l g ∥ L p 3 ∥ h ∥ L p 4

and the product estimates

(2.4) ∥ ∇ l ( g h ) ∥ L p 0 ≲ ∥ g ∥ L p 1 ∥ ∇ l h ∥ L p 2 + ∥ ∇ l g ∥ L p 3 ∥ h ∥ L p 4

hold. In the above, p 0 , p 2 , p 3 ∈ ( 1 , + ∞ ) such that

1 p 0 = 1 p 1 + 1 p 2 = 1 p 3 + 1 p 4 .

Proof

See Lemma 3.1 in the study by Ju [29].□

Lemma 2.3

We have that for l ⩾ 1 ,

(2.5) ∥ ∇ l ℛ ∥ L 2 ≲ δ ( ∥ ∇ n ∥ L 2 + ∥ ∇ l n ∥ L 2 )

and

(2.6) ∥ ∇ l ℛ ∥ L 2 ≲ δ ( ∥ n ∥ L ∞ + ∥ ∇ l n ∥ L 2 ) .

Proof

One can see Lemma A.3 in the study by Tan et al. [48].□

The following lemma has been proved in the study by Tan et al. [47]. We recall the lemma as follows:

Lemma 2.4

Assume that ∥ n ∥ L ∞ ⩽ 1 and ∥ θ ∥ L ∞ ⩽ 1 . Let g ( n , θ ) be a smooth function of n , θ with bounded derivatives of any order, then for any integer k ⩾ 1 and 2 ⩽ p ⩽ ∞ , we have

(2.7) ∥ ∇ k ( g ( n , θ ) ) ∥ L p ≲ ∥ ∇ k n ∥ L p + ∥ ∇ k θ ∥ L p .

3 Global existence of the classical solution

It is well known that the global classical solution of the system (1.1) can follow from the existence and the uniform estimates of the local solution [40]. For the reason that the compressible Navier-Stokes-Smoluchowski equations (1.1) can be reduced to a symmetrizable hyperbolic-parabolic one, according to [30], the local existence of the classical solution is standard. In view of this, it suffices to establish the uniform energy estimates of the Navier-Stokes-Smoluchowski equations.

3.1 Energy estimates

In this subsection, we are devoted to establishing the energy estimates of the local solution ( n , u , η ) T . First of all, under the assumption that the a priori estimates

(3.1) ∥ ( n , u , η ) ( t ) ∥ H k 2 ⩽ δ ,

hold for sufficiently small δ > 0 , the dissipation estimates of the velocity u can be established as follows.

Lemma 3.1

Under the assumption of equation (3.1), it holds that

(3.2) d d t ∫ ( h ′ ( ρ s ) n 2 + ρ s ∣ u ∣ 2 ) d x + ∥ ∇ u ∥ L 2 2 ≲ ∥ η ∥ L 2 2 + δ ∥ ∇ ( n , η ) ∥ L 2 2 ,

and for l = 1 , 2 , … , k , one can obtain

(3.3) d d t ∫ ( h ′ ( ρ s ) ∣ ∇ l n ∣ 2 + ρ s ∣ ∇ l u ∣ 2 ) d x + ∥ ∇ l + 1 u ∥ L 2 2 ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ ( n , η ) ∥ L 2 2 ) + ∥ ∇ l + 1 η ∥ L 2 2 + ∥ ∇ u ∥ L 2 2 .

Proof

Applying the operator ∇ l with 0 ⩽ l ⩽ k to first and second equation in equation (1.7), and taking the inner product with h ′ ( ρ s ) ∇ l n and ρ s ∇ l u , respectively, we can obtain the following identity directly:

(3.4) 1 2 d d t ∫ h ′ ( ρ s ) ∣ ∇ l n ∣ 2 + ρ s ∣ ∇ l u ∣ 2 d x + ∫ h ′ ( ρ s ) ∇ l n ∇ l div ( ρ s u ) + ρ s ∇ l u ⋅ ∇ l ∇ ( h ′ ( ρ s ) n ) d x − ∫ ρ s ∇ l u ⋅ ∇ l 1 ρ s ( μ Δ u + ( μ + λ ) ∇ div u ) d x + ∫ ∇ l ∇ η ρ s ⋅ ρ s ∇ l u d x = ∫ h ′ ( ρ s ) ∇ l n ⋅ ∇ l F 1 d x + ∫ ρ s ∇ l u ⋅ ∇ l F 2 d x .

If l = 0 , by cancellation of the second term on the left-hand side of the identity (3.4), it shows that

(3.5) 1 2 d d t ∫ ( h ′ ( ρ s ) n 2 + ρ s ∣ u ∣ 2 ) d x + μ ∥ ∇ u ∥ L 2 2 + ( μ + λ ) ∥ div u ∥ L 2 2 + ∫ ∇ η ⋅ u d x = ∫ h ′ ( ρ s ) n ⋅ F 1 + ρ s u ⋅ F 2 d x .

By Hölder’s inequality and the interpolation inequality, we can estimate

(3.6) ∫ h ′ ( ρ s ) n ⋅ F 1 d x ≲ ∥ n ∥ L 3 ∥ ∇ u ∥ L 2 ∥ n ∥ L 6 + ∥ n ∥ L 3 ∥ ∇ n ∥ L 2 ∥ u ∥ L 6 ≲ δ ∥ ∇ ( n , u ) ∥ L 2 2 .

The following inequality is a simple consequence of the Hölder’s inequality and Young’s inequality

(3.7) − ∫ ( u ⋅ ∇ u ) ⋅ ρ s u d x − ∫ ρ s u ⋅ ∇ ℛ d x + ∫ 1 ρ s − 1 n + ρ s ∇ η ⋅ ρ s u d x − ∫ η ∇ Φ n + ρ s ⋅ ρ s u d x ≲ ∥ ( n , u ) ∥ L 3 ∥ ∇ ( n , u , η ) ∥ L 2 ∥ u ∥ L 6 + ∥ η ∥ L 2 ∥ ∇ Φ ∥ L 3 ∥ u ∥ L 6 ≲ δ ∥ ∇ ( n , u , η ) ∥ L 2 2 + C ε ∥ η ∥ L 2 2 + ε ∥ ∇ u ∥ L 2 2 ≲ δ ∥ ∇ ( n , u , η ) ∥ L 2 2 + ∥ η ∥ L 2 2 .

It follows from the integration by parts and the Hölder’s inequality that

− ∫ n n + ρ s Δ u ⋅ u d x = ∫ ∇ n n + ρ s ∇ u ⋅ u d x + ∫ n n + ρ s ∇ u ⋅ ∇ u d x ≲ ∇ n n + ρ s L 3 ∥ ∇ u ∥ L 2 ∥ u ∥ L 6 + ∥ n ∥ L ∞ ∥ ∇ u ∥ L 2 ∥ ∇ u ∥ L 2 ≲ δ ∥ ∇ u ∥ L 2 2 .

Direct calculations lead to

(3.8) − ∫ ∇ η ⋅ u d x = ∫ η div u d x ≲ ∥ η ∥ L 2 ∥ ∇ u ∥ L 2 ≲ C ε ∥ η ∥ L 2 2 + ε ∥ ∇ u ∥ L 2 2 .

Therefore, the inequality (3.2) follows from estimates (3.6)–(3.8).

It remains to deal with the case 1 ⩽ l ⩽ k . By employing the commutator notation (2.2) and the integration by parts, the second term on the left-hand side of the identity (3.4) can be reformulated into

∫ h ′ ( ρ s ) ∇ l n ∇ l div ( ρ s u ) + ρ s ∇ l u ⋅ ∇ l ∇ ( h ′ ( ρ s ) n ) d x = ∫ h ′ ( ρ s ) ∇ l n ∇ l div ( ρ s u ) − div ( ρ s ∇ l u ) ∇ l ( h ′ ( ρ s ) n ) d x = ∫ h ′ ( ρ s ) ∇ l n div ( [ ∇ l , ρ s ] u ) + ρ s ∇ l u ⋅ ∇ ( [ ∇ l , h ′ ( ρ s ) ] n ) d x = ∫ h ′ ( ρ s ) ∇ l n ( [ ∇ l , ρ s ] div u + [ ∇ l , ∇ ρ s ] u ) d x + ∫ ρ s ∇ l u ⋅ ( [ ∇ l , h ′ ( ρ s ) ] ∇ n + [ ∇ l , ∇ h ′ ( ρ s ) ] n ) d x .

We obtain after the Hölder’s inequality and the commutator estimates (2.3) that for l = 1 ,

(3.9) ∫ h ′ ( ρ s ) ∇ n ( [ ∇ , ρ s ] div u + [ ∇ , ∇ ρ s ] u ) d x + ∫ ρ s ∇ u ⋅ ( [ ∇ , h ′ ( ρ s ) ] ∇ n + [ ∇ , ∇ h ′ ( ρ s ) ] n ) d x ≲ ∥ ∇ n ∥ L 2 ( ∥ [ ∇ , ρ s ] div u ∥ L 2 + ∥ [ ∇ , ∇ ρ s ] u ∥ L 2 ) + ∥ ∇ u ∥ L 2 ( ∥ [ ∇ , h ′ ( ρ s ) ] ∇ n ∥ L 2 + ∥ [ ∇ , ∇ h ′ ( ρ s ) ] n ∥ L 2 ) ≲ ∥ ∇ n ∥ L 2 ∥ ∇ u ∥ L 2 ⩽ ε ∥ ∇ n ∥ L 2 2 + C ε ∥ ∇ u ∥ L 2 2 ,

where the Young’s inequality is also used. It follows from the commutator estimates (2.3), the Gagliardo-Nirenberg’s inequality (2.1), and the Young’s inequality that for 2 ⩽ l ⩽ k ,

(3.10) ∫ h ′ ( ρ s ) ∇ l n ( [ ∇ l , ρ s ] div u + [ ∇ l , ∇ ρ s ] u ) d x ≲ ∥ ∇ l n ∥ L 2 ( ∥ [ ∇ l , ρ s ] div u ∥ L 2 + ∥ [ ∇ l , ∇ ρ s ] u ∥ L 2 ) ≲ ∥ ∇ l n ∥ L 2 ( ∥ ∇ ρ s ∥ L ∞ ∥ ∇ l − 1 div u ∥ L 2 + ∥ ∇ l ρ s ∥ L 6 ∥ div u ∥ L 3 ) + ∥ ∇ l n ∥ L 2 ( ∥ ∇ 2 ρ s ∥ L 3 ∥ ∇ l − 1 u ∥ L 6 + ∥ ∇ l + 1 ρ s ∥ L 2 ∥ u ∥ L ∞ ) ≲ ∥ ∇ l n ∥ L 2 ( ∥ ∇ l u ∥ L 2 + ∥ ∇ u ∥ L 3 + ∥ u ∥ L ∞ ) ⩽ ε ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 ) + C ε ∥ ∇ u ∥ L 2 2

and

(3.11) ∫ ρ s ∇ l u ⋅ ( [ ∇ l , h ′ ( ρ s ) ] ∇ n + [ ∇ l , ∇ h ′ ( ρ s ) ] n ) d x ≲ ∥ ∇ l u ∥ L 2 ∥ [ ∇ l , h ′ ( ρ s ) ] ∇ n ∥ L 2 + ∥ ∇ l u ∥ L 2 ∥ [ ∇ l , ∇ h ′ ( ρ s ) ] n ∥ L 2 ≲ ∥ ∇ l u ∥ L 2 ( ∥ ∇ h ′ ( ρ s ) ∥ L ∞ ∥ ∇ l n ∥ L 2 + ∥ ∇ l h ′ ( ρ s ) ∥ L 6 ∥ ∇ n ∥ L 3 ) + ∥ ∇ l u ∥ L 2 ( ∥ ∇ 2 h ′ ( ρ s ) ∥ L 3 ∥ ∇ l − 1 n ∥ L 6 + ∥ ∇ l + 1 h ′ ( ρ s ) ∥ L 2 ∥ n ∥ L ∞ ) ≲ ∥ ∇ l u ∥ L 2 ( ∥ ∇ l n ∥ L 2 + ∥ ∇ n ∥ L 3 + ∥ n ∥ L ∞ ) ≲ ε ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ n ∥ L 2 2 ) + ∥ ∇ u ∥ L 2 2 .

In terms of estimates (3.9)–(3.11), it implies that

(3.12) ∫ h ′ ( ρ s ) ∇ l n ∇ l div ( ρ s u ) + ρ s ∇ l u ⋅ ∇ l ∇ ( h ′ ( ρ s ) n ) d x ≲ ε ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ n ∥ L 2 2 ) + ∥ ∇ u ∥ L 2 2 .

We can employ the integration by parts and the commutator notation (2.2) to rewrite

− ∫ ρ s ∇ l u ⋅ ∇ l 1 ρ s ∇ div u d x = − ∫ ρ s ∇ l u ⋅ ∇ l ∇ 1 ρ s div u − ∇ 1 ρ s div u d x = ∫ div ( ρ s ∇ l u ) ∇ l 1 ρ s div u + ρ s ∇ l u ⋅ ∇ l ∇ 1 ρ s div u d x = ∫ ∣ ∇ l div u ∣ 2 d x + ∫ ρ s ∇ l div u ⋅ ∇ l , 1 ρ s div u d x + ∫ ∇ ρ s ⋅ ∇ l u ∇ l 1 ρ s div u d x + ∫ ρ s ∇ l u ⋅ ∇ l ∇ 1 ρ s div u d x .

It infers from the Hölder’s inequality and the product estimates (2.4) that for l = 1 ,

(3.13) ∫ ρ s ∇ div u ⋅ ∇ , 1 ρ s div u d x + ∫ ∇ ρ s ⋅ ∇ u ∇ 1 ρ s div u d x ≲ ∥ ∇ div u ∥ L 2 ∇ , 1 ρ s div u L 2 + ∥ ∇ u ∥ L 2 ∇ 1 ρ s div u L 2 ≲ ∥ ∇ 2 u ∥ L 2 ∇ 1 ρ s L ∞ ∥ div u ∥ L 2 + ∥ ∇ u ∥ L 2 1 ρ s L ∞ ∥ ∇ div u ∥ L 2 + ∇ 1 ρ s L 3 ∥ div u ∥ L 6 ≲ ∥ ∇ 2 u ∥ L 2 ∥ ∇ u ∥ L 2 ⩽ ε ∥ ∇ 2 u ∥ L 2 2 + C ε ∥ ∇ u ∥ L 2 2

and

(3.14) ∫ ρ s ∇ u ⋅ ∇ ∇ 1 ρ s div u d x ≲ ∥ ∇ u ∥ L 2 ∇ ∇ 1 ρ s div u L 2 ≲ ∥ ∇ u ∥ L 2 ∇ 1 ρ s L ∞ ∥ ∇ div u ∥ L 2 + ∇ 2 1 ρ s L 3 ∥ div u ∥ L 6 ≲ ∥ ∇ 2 u ∥ L 2 ∥ ∇ u ∥ L 2 ≲ ε ∥ ∇ 2 u ∥ L 2 2 + C ε ∥ ∇ u ∥ L 2 2 .

We employ the Hölder’s inequality, the commutator estimates (2.3) along with the Gagliardo-Nirenberg’s inequality (2.1), and the Young’s inequality to estimate, for 2 ⩽ l ⩽ k :

(3.15) ∫ ρ s ∇ l div u ∇ l , 1 ρ s div u d x ≲ ∥ ∇ l div u ∥ L 2 ∇ l , 1 ρ s div u L 2 ≲ ∥ ∇ l + 1 u ∥ L 2 ∇ 1 ρ s L ∞ ∥ ∇ l u ∥ L 2 + ∇ l 1 ρ s L 6 ∥ div u ∥ L 3 ≲ ∥ ∇ l + 1 u ∥ L 2 ( ∥ ∇ l u ∥ L 2 + ∥ ∇ u ∥ L 3 ) ≲ ∥ ∇ l + 1 u ∥ L 2 ( ∥ ∇ l u ∥ L 2 + ∥ ∇ u ∥ L 2 ) ≲ ε ∥ ∇ l + 1 u ∥ L 2 2 + C ε ∥ ∇ u ∥ L 2 2 .

Combining the product estimates (2.4) and the Gagliardo-Nirenberg’s inequality (2.1), due to the Young’s inequality, for 2 ⩽ l ⩽ k , it gives

(3.16) ∫ ∇ ρ s ⋅ ∇ l u ∇ l 1 ρ s div u d x ≲ ∥ ∇ l u ∥ L 2 ∇ l 1 ρ s div u L 2 ≲ ∥ ∇ l u ∥ L 2 1 ρ s L ∞ ∥ ∇ l div u ∥ L 2 + ∇ l 1 ρ s L 6 ∥ div u ∥ L 3 ≲ ∥ ∇ l u ∥ L 2 ( ∥ ∇ l + 1 u ∥ L 2 + ∥ ∇ u ∥ L 3 ) ≲ ∥ ∇ l u ∥ L 2 ( ∥ ∇ l + 1 u ∥ L 2 + ∥ ∇ u ∥ L 2 ) ≲ ε ∥ ∇ l + 1 u ∥ L 2 2 + C ε ∥ ∇ u ∥ L 2 2

and

(3.17) ∫ ρ s ∇ l u ⋅ ∇ l ∇ 1 ρ s div u d x ≲ ∥ ∇ l u ∥ L 2 ∇ l ∇ 1 ρ s div u L 2 ≲ ∥ ∇ l u ∥ L 2 ∇ 1 ρ s L ∞ ∥ ∇ l div u ∥ L 2 + ∇ l + 1 1 ρ s L 2 ∥ div u ∥ L ∞ ≲ ∥ ∇ l u ∥ L 2 ( ∥ ∇ l + 1 u ∥ L 2 + ∥ ∇ u ∥ L ∞ ) ≲ ∥ ∇ l u ∥ L 2 ( ∥ ∇ l + 1 u ∥ L 2 + ∥ ∇ u ∥ L 2 ) ≲ ε ∥ ∇ l + 1 u ∥ L 2 2 + C ε ∥ ∇ u ∥ L 2 2 .

In terms of estimates (3.13)–(3.17), it implies that

(3.18) − ∫ ρ s ∇ l u ⋅ ∇ l 1 ρ s ∇ div u d x ⩾ ∥ ∇ l + 1 u ∥ L 2 2 − ( ε ∥ ∇ l + 1 u ∥ L 2 2 + C ε ∥ ∇ u ∥ L 2 2 )

and, similarly,

(3.19) − ∫ ρ s ∇ l u ⋅ ∇ l 1 ρ s Δ u d x ⩾ ∥ ∇ l + 1 u ∥ L 2 2 − ( ε ∥ ∇ l + 1 u ∥ L 2 2 + C ε ∥ ∇ u ∥ L 2 2 ) .

From the product estimates (2.4), the Gagliardo-Nirenberg’s inequality (2.1), and the Young’s inequality, one can conclude that

(3.20) − ∫ ∇ l ∇ η ρ s ⋅ ρ s ∇ l u d x ≲ ∇ l ∇ η ρ s L 2 ∥ ∇ l u ∥ L 2 ≲ ( ∥ ∇ l + 1 η ∥ L 2 + ∥ ∇ l ρ s ∥ L ∞ ∥ ∇ η ∥ L 2 ) ∥ ∇ l u ∥ L 2 ≲ ∥ ∇ l + 1 η ∥ L 2 2 + ε ∥ ∇ η ∥ L 2 2 + ∥ ∇ l u ∥ L 2 2 ≲ ∥ ∇ l + 1 η ∥ L 2 2 + ε ∥ ∇ η ∥ L 2 2 + ε ∥ ∇ l + 1 u ∥ L 2 2 + C ε ∥ ∇ u ∥ L 2 2 .

Now, we turn to estimate the term

∫ h ′ ( ρ s ) ∇ l n ⋅ ∇ l F 1 d x .

We use the commutator notation (2.2) to rewrite

∫ h ′ ( ρ s ) ∇ l n ∇ l div ( n u ) d x = − ∫ h ′ ( ρ s ) ∇ l n ∇ l ( u ⋅ ∇ n + n div u ) d x = − ∫ h ′ ( ρ s ) u ⋅ ∇ ∇ l n ∇ l n d x − ∫ h ′ ( ρ s ) ∇ l n [ ∇ l , u ] ⋅ ∇ n d x − ∫ h ′ ( ρ s ) ∇ l n ∇ l ( n div u ) d x = I 11 + I 12 + I 13 .

It infers from the integration by parts and the Hölder’s inequality that

(3.21) I 11 = − 1 2 ∫ h ′ ( ρ s ) u ⋅ ∇ ∣ ∇ l n ∣ 2 d x = 1 2 ∫ div ( h ′ ( ρ s ) u ) ∣ ∇ l n ∣ 2 d x ≲ ( ∥ u ∥ L ∞ + ∥ div u ∥ L ∞ ) ∥ ∇ l n ∥ L 2 2 ≲ δ ∥ ∇ l n ∥ L 2 2 .

By the application of the interpolation inequality (2.1) and the commutator estimates (2.3), we can deduce that

(3.22) I 12 ≲ ∥ [ ∇ l , u ] ⋅ ∇ n ∥ L 2 ∥ ∇ l n ∥ L 2 ≲ ( ∥ ∇ u ∥ L ∞ ∥ ∇ l n ∥ L 2 + ∥ ∇ l u ∥ L 6 ∥ ∇ n ∥ L 3 ) ∥ ∇ l n ∥ L 2 ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 ) .

It implies from the product estimates (2.4) that

(3.23) I 13 ≲ ∥ ∇ l ( n div u ) ∥ L 2 ∥ ∇ l n ∥ L 2 ≲ ( ∥ n ∥ L ∞ ∥ ∇ l div u ∥ L 2 + ∥ ∇ l n ∥ L 2 ∥ div u ∥ L ∞ ) ∥ ∇ l n ∥ L 2 ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 ) .

In terms of estimates (3.21)–(3.23), the term ∫ h ′ ( ρ s ) ∇ l n ∇ l div ( n u ) d x can be controlled as follows:

(3.24) ∫ h ′ ( ρ s ) ∇ l n ∇ l div ( n u ) d x ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 ) .

It is derived from the product estimates (2.4) along with the Young’s inequality that

(3.25) − ∫ ρ s ∇ l u ⋅ ∇ l ( u ⋅ ∇ u ) d x ≲ ∥ ∇ l ( u ⋅ ∇ u ) ∥ L 2 ∥ ∇ l u ∥ L 2 ≲ ( ∥ u ∥ L ∞ ∥ ∇ l ∇ u ∥ L 2 + ∥ ∇ l u ∥ L 2 ∥ ∇ u ∥ L ∞ ) ∥ ∇ l u ∥ L 2 ≲ δ ( ∥ ∇ l u ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 ) ≲ δ ( ∥ ∇ u ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 ) .

We can deduce from equation (2.5) of Lemma 2.3 that

(3.26) − ∫ ρ s ∇ l u ⋅ ∇ l ∇ ℛ d x = ∫ ∇ ρ s ⋅ ∇ l u ⋅ ∇ l ℛ d x + ∫ ρ s ∇ l div u ⋅ ∇ l ℛ d x ≲ ∥ ∇ ρ s ∥ L 3 ∥ ∇ l u ∥ L 6 ∥ ∇ l ℛ ∥ L 2 + ∥ ∇ l + 1 u ∥ L 2 ∥ ∇ l ℛ ∥ L 2 ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ n ∥ L 2 2 ) .

We should distinguish the order l while estimating the following term:

∫ ρ s ∇ l u ⋅ ∇ l 1 ρ s − 1 n + ρ s Δ u d x .

If l = 1 , it obviously holds that

(3.27) ∫ ρ s ∇ u ⋅ ∇ 1 ρ s − 1 n + ρ s Δ u d x = − ∫ ( ∇ ρ s ∇ u + ρ s ∇ 2 u ) ⋅ 1 ρ s − 1 n + ρ s Δ u d x ≲ ∥ ∇ u ∥ L 6 1 ρ s − 1 n + ρ s L 3 ∥ ∇ 2 u ∥ L 2 + ∥ ∇ 2 u ∥ L 2 1 ρ s − 1 n + ρ s L ∞ ∥ ∇ 2 u ∥ L 2 ≲ δ ∥ ∇ 2 u ∥ L 2 2 .

If 2 ⩽ l ⩽ k , by employing the integration by parts and the product estimates (2.4), it can be estimated as follows:

(3.28) ∫ ρ s ∇ l u ⋅ ∇ l 1 ρ s − 1 n + ρ s Δ u d x = − ∫ ∇ ρ s ∇ l u ⋅ ∇ l − 1 1 ρ s − 1 n + ρ s Δ u + ρ s ∇ l + 1 u ⋅ ∇ l − 1 1 ρ s − 1 n + ρ s Δ u d x ≲ ∥ ∇ l u ∥ L 6 ∇ l − 1 1 ρ s − 1 n + ρ s Δ u L 6 ∕ 5 + ∥ ∇ l + 1 u ∥ L 2 ∇ l − 1 1 ρ s − 1 n + ρ s Δ u L 2 ≲ ∥ ∇ l + 1 u ∥ L 2 ∇ l − 1 1 ρ s − 1 n + ρ s L 2 ∥ ∇ 2 u ∥ L 3 + 1 ρ s − 1 n + ρ s L 3 ∥ ∇ l + 1 u ∥ L 2 + ∥ ∇ l + 1 u ∥ L 2 ∇ l − 1 1 ρ s − 1 n + ρ s L 6 ∥ ∇ 2 u ∥ L 3 + 1 ρ s − 1 n + ρ s L ∞ ∥ ∇ l + 1 u ∥ L 2 ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ l − 1 n ∥ L 2 2 ) + ∥ ∇ l + 1 u ∥ L 2 ∥ ∇ 2 u ∥ L 3 ⩽ ( δ + ε ) ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ n ∥ L 2 2 ) + C ε ∥ ∇ u ∥ L 2 2 .

From estimates (3.27) and (3.28), we have

(3.29) ∫ ρ s ∇ l u ⋅ ∇ l 1 ρ s − 1 n + ρ s Δ u d x ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ n ∥ L 2 2 ) + ∥ ∇ u ∥ L 2 2 .

Similarly to equation (3.29), it holds

(3.30) ∫ ρ s ∇ l u ⋅ ∇ l 1 ρ s − 1 n + ρ s ∇ div u d x ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ n ∥ L 2 2 ) + ∥ ∇ u ∥ L 2 2 .

One can deduce from the Hölder’s inequality and the product estimates (2.4) that

(3.31) ∫ ∇ l η ∇ Φ n + ρ s ⋅ ρ s ∇ l u d x ≲ ∇ l η ∇ Φ n + ρ s L 2 ∥ ∇ l u ∥ L 2 ≲ ∥ η ∥ L ∞ ∇ l ∇ Φ n + ρ s L 2 + ∥ ∇ l η ∥ L 6 ∇ Φ n + ρ s L 3 ∥ ∇ l u ∥ L 2 ≲ ( ∥ η ∥ L ∞ + ∥ η ∥ L ∞ ∥ ∇ l n ∥ L 2 + ∥ ∇ l + 1 η ∥ L 2 ) ∥ ∇ l u ∥ L 2 ≲ ( ∥ η ∥ L ∞ + δ ∥ ∇ l n ∥ L 2 + ∥ ∇ l + 1 η ∥ L 2 ) ∥ ∇ l u ∥ L 2 ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ η ∥ 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ l + 1 η ∥ L 2 2 ) + ∥ ∇ u ∥ L 2 2

and

(3.32) ∫ ∇ l 1 n + ρ s − 1 ρ s ∇ η ⋅ ρ s ∇ l u d x ≲ ∇ l 1 n + ρ s − 1 ρ s ∇ η L 2 ∥ ∇ l u ∥ L 2 ≲ 1 n + ρ s − 1 ρ s L ∞ ∥ ∇ l + 1 η ∥ L 2 + ∇ l 1 n + ρ s − 1 ρ s L 2 ∥ ∇ η ∥ L ∞ ∥ ∇ l u ∥ L 2 ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ η ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ l + 1 η ∥ L 2 2 ) + ∥ ∇ u ∥ L 2 2 ,

where the Young’s inequality is also be used. To this end, in light of estimate (3.12) and estimates (3.18)–(3.20), along with estimates (3.24)–(3.26) and estimates (3.29)–(3.32), we can prove estimates (3.3) to be true.□

Second, we derive the dissipation estimates for η . In order to close the energy estimates, it is necessary to distinguish the energy estimates of η by the value of order l .

Lemma 3.2

It holds for l = 0

(3.33) d d t ∫ ∣ η ∣ 2 d x + ∥ ∇ η ∥ L 2 2 ≲ ∥ η ∥ L 2 2 ,

and for 1 ⩽ l ⩽ k ,

(3.34) d d t ∫ ∣ ∇ l η ∣ 2 d x + ∥ ∇ l + 1 η ∥ L 2 2 ≲ ∥ ∇ η ∥ L 2 2 + δ ∥ ∇ l + 1 u ∥ L 2 2 .

Proof

Taking the inner product of the third equation in equation (1.7) with η , together with the integration by parts, we obtain

1 2 d d t ∫ ∣ η ∣ 2 d x + ∥ ∇ η ∥ L 2 2 = ∫ η u ⋅ ∇ η − η ∇ Φ ⋅ ∇ η d x ≲ ∥ ∇ η ∥ L 2 ∥ η ∥ L 6 ∥ u ∥ L 3 + ∥ ∇ Φ ∥ L ∞ ∥ η ∥ L 2 ∥ ∇ η ∥ L 2 ≲ δ ∥ ∇ η ∥ L 2 2 + ε ∥ ∇ η ∥ L 2 2 + C ε ∥ η ∥ L 2 2 ,

which implies equation (3.33), since δ and ε are sufficiently small. The energy estimate (3.34) at l level with 1 ⩽ l ⩽ k can be obtained as follows:

□ 1 2 d d t ∫ ∣ ∇ l η ∣ 2 d x + ∥ ∇ l + 1 η ∥ L 2 2 = − ∫ ∇ l ( η ∇ Φ ) ⋅ ∇ l + 1 η d x + ∫ ∇ l ( η u ) ⋅ ∇ l + 1 η d x ≲ ∥ ∇ l ( η ∇ Φ ) ∥ L 2 ∥ ∇ l + 1 η ∥ L 2 + ∥ ∇ l ( η u ) ∥ L 2 ∥ ∇ l + 1 η ∥ L 2 ≲ ( ∥ ∇ l η ∥ L 2 ∥ ∇ Φ ∥ L ∞ + ∥ η ∥ L ∞ ∥ ∇ l ∇ Φ ∥ L 2 ) ∥ ∇ l + 1 η ∥ L 2 + δ ∥ ∇ l + 1 ( u , η ) ∥ L 2 2 ≲ ∥ ∇ η ∥ L 2 2 + δ ∥ ∇ l + 1 ( u , η ) ∥ L 2 2 .

The energy estimates established in Lemmas 3.1–3.2 are not sufficient to close the energy estimates due to the lack of the dissipation estimates of n . We can use the second equation in equation (1.7) to obtain the dissipation estimates of n as follows.

Lemma 3.3

Under the assumption of equation (3.1), we can deduce

(3.35) d d t ∫ u ⋅ ∇ ( h ′ ( ρ s ) n ) d x + ∥ ∇ n ∥ L 2 2 ≲ ∥ ∇ 2 u ∥ L 2 2 + ∥ ∇ ( u , η ) ∥ L 2 2 .

Proof

Taking the inner product of the second equation in equation (1.7) with ∇ ( h ′ ( ρ s ) n ) gives

(3.36) ∫ ∂ t u ⋅ ∇ ( h ′ ( ρ s ) n ) d x + ∥ ∇ ( h ′ ( ρ s ) n ) ∥ L 2 2 ≲ ∥ ∇ 2 u ∥ L 2 2 + ∥ ∇ η ∥ L 2 2 + ∥ u ⋅ ∇ u ∥ L 2 2 + ∥ ∇ ℛ ∥ L 2 2 + 1 ρ s − 1 n + ρ s L ∞ 2 ∥ ∇ 2 u ∥ L 2 2 + 1 ρ s − 1 n + ρ s L ∞ 2 ∥ ∇ η ∥ L 2 2 + η ∇ Φ n + ρ s L 2 2 ≲ ∥ ∇ 2 u ∥ L 2 2 + ∥ ∇ η ∥ L 2 2 + δ ∥ ∇ ( n , u ) ∥ L 2 2 + ∥ n ∥ L ∞ 2 ∥ ( ∇ 2 u , ∇ η ) ∥ L 2 2 + ∥ η ∥ L 6 2 ∥ ∇ Φ ∥ L 3 2 ≲ ∥ ∇ 2 u ∥ L 2 2 + ∥ ∇ η ∥ L 2 2 + δ ∥ ∇ ( n , u , η ) ∥ L 2 2 .

For the first term of the inequality (3.36), by integrating by parts and making use of the first equation of (1.7), it is obvious to have

(3.37) − ∫ ∂ t u ⋅ ∇ ( h ′ ( ρ s ) n ) d x = − d d t ∫ u ⋅ ∇ ( h ′ ( ρ s ) n ) d x + ∫ u ⋅ ∇ ( h ′ ( ρ s ) ∂ t n ) d x = − d d t ∫ u ⋅ ∇ ( h ′ ( ρ s ) n ) d x + ∫ h ′ ( ρ s ) div u ( u ⋅ ∇ n + n div u + ρ s div u + u ⋅ ∇ ρ s ) d x ⩽ − d d t ∫ u ⋅ ∇ ( h ′ ( ρ s ) n ) d x + C ∥ ∇ u ∥ L 2 2 + δ ∥ ∇ n ∥ L 2 2 .

The combination of estimates (3.36) and (3.37) leads to equation (3.35).□

In a similar way, the dissipation estimates of n at l level with 1 ⩽ l ⩽ k − 1 can also be obtained.

Lemma 3.4

For integer l with 1 ⩽ l ⩽ k − 1 , under the assumption of equation (3.1), we conclude

(3.38) d d t ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∥ ∇ l + 1 n ∥ L 2 2 ≲ ∥ ∇ l + 2 u ∥ L 2 2 + ∥ ∇ l + 1 ( u , η ) ∥ L 2 2 + ∥ ∇ ( n , u , η ) ∥ L 2 2 .

Proof

Applying ∇ l with 1 ⩽ l ⩽ k − 1 to (1.7)₂ and then taking the L 2 inner product with ∇ ∇ l n , it gives

(3.39) ∫ ∇ l ∂ t u ⋅ ∇ l ∇ n d x + ∫ h ′ ( ρ s ) ∣ ∇ l + 1 n ∣ 2 d x ≲ ( ∥ [ ∇ l + 1 , h ′ ( ρ s ) ] n ∥ L 2 + ∥ ∇ l ( u ⋅ ∇ u ) ∥ L 2 + ∥ ∇ l + 1 ℛ ∥ L 2 + ∇ l 1 n + ρ s Δ u L 2 + ∇ l 1 n + ρ s ∇ η L 2 + ∇ l η ∇ Φ n + ρ s L 2 ∥ ∇ l + 1 n ∥ L 2 .

We can take a similar approach to equation (3.37), using the Hölder’s inequality and the product estimates (2.4), to obtain

(3.40) − ∫ ∇ l ∂ t u ⋅ ∇ ∇ l n d x = − d d t ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∫ ∇ l u ⋅ ∇ ∇ l ∂ t n d x = − d d t ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∫ ∇ l div u ∇ l ( u ⋅ ∇ n + n div u + ρ s div u + u ⋅ ∇ ρ s ) d x ⩽ − d d t ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ l ( u ⋅ ∇ n ) ∥ L 2 2 + ∥ ∇ l ( n div u ) ∥ L 2 2 + ∥ ∇ l ( ρ s div u ) ∥ L 2 2 + ∥ ∇ l ( u ⋅ ∇ ρ s ) ∥ L 2 2 ⩽ − d d t ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ u ∥ L 2 2 + δ ∥ ∇ l + 1 n ∥ L 2 2 .

By the commutator estimates (2.3), it infers

(3.41) ∥ [ ∇ l + 1 , h ′ ( ρ s ) ] n ∥ L 2 ≲ ∥ ∇ h ′ ( ρ s ) ∥ L ∞ ∥ ∇ l n ∥ L 2 + ∥ ∇ l + 1 h ′ ( ρ s ) ∥ L 3 ∥ n ∥ L 6 ≲ ∥ ∇ l n ∥ L 2 + ∥ ∇ n ∥ L 2 ⩽ ε ∥ ∇ l + 1 n ∥ L 2 + C ε ∥ ∇ n ∥ L 2 .

With the help of the product estimates (2.4) and the Young’s inequality, we deduce

(3.42) ∇ l 1 n + ρ s Δ u L 2 ≲ 1 n + ρ s L ∞ ∥ ∇ l + 2 u ∥ L 2 + ∇ l 1 n + ρ s L 6 ∥ ∇ 2 u ∥ L 3 ≲ ∥ ∇ l + 2 u ∥ L 2 + ∥ ∇ l + 1 n ∥ L 2 ∥ ∇ 2 u ∥ L 3 + ∥ ∇ 2 u ∥ L 3 ≲ ∥ ∇ l + 2 u ∥ L 2 + ∥ ∇ u ∥ L 2 + ε ∥ ∇ l + 1 n ∥ L 2

and

(3.43) ∇ l 1 n + ρ s ∇ η L 2 ≲ 1 n + ρ s L ∞ ∥ ∇ l + 1 η ∥ L 2 + ∥ ∇ l ( n , ρ s ) ∥ L 6 ∥ ∇ η ∥ L 3 ≲ ∥ ∇ l + 1 η ∥ L 2 + ∥ ∇ l + 1 n ∥ L 2 ∥ ∇ η ∥ L 3 + ∥ ∇ η ∥ L 3 ≲ ∥ ∇ l + 1 η ∥ L 2 + ∥ ∇ η ∥ L 2 + ε ∥ ∇ l + 1 n ∥ L 2 .

We can adopt a similar approach to equations (3.42) and (3.43) to have

(3.44) ∇ l η ∇ Φ n + ρ s L 2 ≲ ∥ η ∥ L ∞ ∇ l ∇ Φ n + ρ s L 2 + ∥ ∇ l η ∥ L 6 ∇ Φ n + ρ s L 3 ≲ δ ∥ ∇ l n ∥ L 2 + ∥ ∇ l + 1 η ∥ L 2 + ∥ η ∥ L ∞ ≲ δ ∥ ∇ l + 1 n ∥ L 2 + ∥ ∇ l + 1 η ∥ L 2 + ∥ ∇ ( n , η ) ∥ L 2 .

The estimate (3.38) follow from equations (3.40)–(3.44).□

Based on Lemmas 3.1–3.4, at this point, it is able to prove Theorem 1.1.

3.2 Proof of Theorem 1.1

The summation of equations (3.2) and (3.3) with l = 1 , 2 , ⋯ , k in Lemma 3.1 gives

(3.45) d d t ( ∥ n ∥ H k 2 + ∥ u ∥ H k 2 ) + ∑ ℓ = 1 ℓ = k + 1 ∥ ∇ ℓ u ∥ L 2 2 ≲ ∥ η ∥ L 2 2 + δ ∑ ℓ = 1 ℓ = k ∥ ∇ ℓ n ∥ L 2 2 + ∑ ℓ = 1 ℓ = k + 1 ∥ ∇ ℓ η ∥ L 2 2 .

Summing up equation (3.35) in Lemma 3.3 and equation (3.38) in Lemma 3.4 from 1 to k − 1 , it infers

(3.46) d d t ∑ ℓ = 0 ℓ = k − 1 ∫ ∇ ℓ u ⋅ ∇ ∇ ℓ n d x + ∑ ℓ = 1 ℓ = k ∥ ∇ ℓ n ∥ L 2 2 ≲ ∑ ℓ = 1 ℓ = k + 1 ∥ ∇ ℓ u ∥ L 2 2 + ∑ ℓ = 1 ℓ = k ∥ ∇ ℓ η ∥ L 2 2 .

It follows from equations (3.33) and (3.34) with 1 ⩽ l ⩽ k in Lemma 3.2 that

(3.47) d d t ∥ η ∥ H k 2 + ∑ ℓ = 1 ℓ = k + 1 ∥ ∇ ℓ η ∥ L 2 2 ≲ ∥ η ∥ L 2 2 + δ ∑ ℓ = 2 ℓ = k + 1 ∥ ∇ ℓ u ∥ L 2 2 .

A suitable linear combination of equations (3.45) and (3.46) implies that

d d t ∥ n ∥ H k 2 + ∥ u ∥ H k 2 + ∑ ℓ = 0 ℓ = k − 1 ∫ ∇ ℓ u ⋅ ∇ ∇ ℓ n d x + ∑ ℓ = 1 ℓ = k ∥ ∇ ℓ n ∥ L 2 2 + ∑ ℓ = 1 ℓ = k + 1 ∥ ∇ ℓ u ∥ L 2 2 ≲ ∥ η ∥ L 2 2 + ∑ ℓ = 1 ℓ = k + 1 ∥ ∇ ℓ η ∥ L 2 2 ,

which combined with equation (3.47) gives

(3.48) d d t ∥ n ∥ H k 2 + ∥ u ∥ H k 2 + ∥ η ∥ H k 2 + ∑ ℓ = 0 ℓ = k − 1 ∫ ∇ ℓ u ⋅ ∇ ∇ ℓ n d x + ∑ ℓ = 1 ℓ = k ∥ ∇ ℓ n ∥ L 2 2 + ∑ ℓ = 1 ℓ = k + 1 ∥ ∇ ℓ u ∥ L 2 2 + ∑ ℓ = 1 ℓ = k + 1 ∥ ∇ ℓ η ∥ L 2 2 ≲ ∥ η ∥ L 2 2 .

In order to obtain the uniform estimates of the solution ( n , u , η ) T , we assume

(3.49) Z ( t ) = sup 0 ⩽ τ ⩽ t ( 1 + τ ) 3 2 1 p − 1 2 ∥ η ∥ L 2

and claim that Z ( t ) ⩽ C . In fact, under the assumption of Theorem 1.1, by the Duhamel principle and making use of the linear decay estimates of the parabolic equation, we have

∥ η ∥ L 2 ⩽ ( 1 + t ) − 3 2 1 p − 1 2 ∥ η 0 ∥ L p + ∫ 0 t ( 1 + t − τ ) − 5 4 ( ∥ η ∥ L 2 ∥ u ∥ L 2 + ∥ η ∥ L 2 ∥ ∇ Φ ∥ L 2 ) d τ ⩽ ( 1 + t ) − 3 2 1 p − 1 2 ∥ η 0 ∥ L p + δ ∫ 0 t ( 1 + t − τ ) − 5 4 ∥ η ∥ L 2 d τ ⩽ ( 1 + t ) − 3 2 1 p − 1 2 ∥ η 0 ∥ L p + δ ∫ 0 t ( 1 + t − τ ) − 5 4 ( 1 + τ ) − 3 2 1 p − 1 2 Z ( t ) d τ ⩽ ( 1 + t ) − 3 2 1 p − 1 2 ∥ η 0 ∥ L p + δ ( 1 + t ) − 3 2 1 p − 1 2 Z ( t ) ,

which along with definition (3.49) gives

Z ( t ) ⩽ ∥ η 0 ∥ L p + δ Z ( t ) .

Since δ is sufficiently small, the term δ Z ( t ) can be absorbed, and we directly deduce

(3.50) ∥ η ∥ L 2 ⩽ C ( 1 + t ) − 3 2 1 p − 1 2 ∥ η 0 ∥ L p .

With the help of equation (3.50) and taking the integration of equation (3.48) about the time variable t , we have

(3.51) ∥ ( n , u , η ) ∥ H k 2 + ∫ 0 t ( ∥ ∇ n ∥ H k − 1 2 + ∥ ∇ ( u , η ) ∥ H k 2 ) d τ ≲ ∥ ( n 0 , u 0 , η 0 ) ∥ H k 2 + ∫ 0 t ∥ η ∥ L 2 2 d τ ≲ ∥ ( n 0 , u 0 , η 0 ) ∥ H k 2 + C ∫ 0 t ( 1 + τ ) − 3 1 p − 1 2 ∥ η 0 ∥ L p 2 d τ ≲ ∥ ( n 0 , u 0 , η 0 ) ∥ H k 2 + ∥ η 0 ∥ L p 2 .

To this end, we have obtained the uniform estimates of the solution and finished the proof of Theorem 1.1.

4 Time decay rates of the solution

This section is devoted to the time decay rates of the solution ( n , u , η ) T . Since the system (1.7) is an evolution equation with variable coefficient, we cannot use the Fourier transform directly to obtain the linear decay property of the solution. Motivated by the work of Tan et al. [48], we reformulate equation (1.7) as follows:

(4.1) ∂ t n + ρ ¯ div u = G 1 , ∂ t u + h ′ ( ρ ¯ ) ∇ n − 1 ρ ¯ ( μ Δ u + ( μ + λ ) ∇ div u ) + 1 ρ ¯ ∇ η = G 2 , ∂ t η − Δ η = F 3 , ( n , u , η ) ∣ t = 0 = ( n 0 , u 0 , η 0 ) ,

with the nonlinear terms G 1 and G 2 denoted by

G 1 = − div ( ( n + ρ s − ρ ¯ ) u ) , G 2 = − u ⋅ ∇ u − ∇ ℛ − ∇ ( ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) n ) − 1 ρ ¯ − 1 n + ρ s ( μ Δ u + ( μ + λ ) ∇ div u ) + 1 ρ ¯ − 1 n + ρ s ∇ η − η ∇ Φ n + ρ s .

By the Duhamel principle, the solution ( n , u , η ) T to equation (4.1) can be expressed as follows:

(4.2) ( n , u , η ) T ( t ) = e − t A ( n 0 , u 0 , η 0 ) T + ∫ 0 t e − ( t − τ ) A ( G 1 , G 2 , F 3 ) T ( τ ) d τ ,

where A is defined by

0 , ρ ¯ div , 0 h ′ ( ρ ¯ ) ∇ , − 1 ρ ¯ ( μ Δ + ( μ + λ ) ∇ div ) , 1 ρ ¯ ∇ 0 , 0 , − Δ .

Based on the spectral analysis in the study by Chen et al. [7], the time decay property of the solution ( n , u , η ) T to equation (4.1) can be shown as follows.

Proposition 4.1

For 1 ⩽ p , r ⩽ 2 , q ⩾ 2 , and ℓ ⩾ 0 , we have

(4.3) ∥ ∇ ℓ ( n , u , η ) ( t ) ∥ L q ≲ ( 1 + t ) − 3 2 1 p − 1 q − ℓ 2 ( ∥ ( n 0 , u 0 , η 0 ) ∥ L p + ∥ ∇ ℓ ( n 0 , u 0 , η 0 ) ∥ L q ) + ∫ 0 t ( 1 + t − τ ) − 3 2 1 r − 1 q − ℓ 2 ( ∥ ( G 1 , G 2 , F 3 ) ( τ ) ∥ L r + ∥ ∇ ℓ ( G 1 , G 2 , F 3 ) ( τ ) ∥ L q ) d τ .

4.1 Energy estimates

We should make a little preparations for the later use in the proof of Theorem 1.3. From Theorem 1.1, it is known that ∥ ( n , u , η ) ( t ) ∥ H k ⩽ δ . Under the further assumption that

(4.4) ∥ ∇ Φ ( t ) ∥ H k + ∥ ∇ Φ ( t ) ∥ L r ⩽ δ

for some small δ > 0 , we can perform the energy method again to derive the energy estimates of ( n , u , η ) T and its higher-order derivative. We first show the energy estimates of ( n , u ) in the following lemma.

Lemma 4.2

Suppose that 0 ⩽ l ⩽ k , and under the assumption of Theorem 1.3, we have

(4.5) d d t ∫ ( h ′ ( ρ ¯ ) ∣ ∇ l n ∣ 2 + ρ ¯ ∣ ∇ l u ∣ 2 ) d x + ∥ ∇ l + 1 u ∥ L 2 2 ≲ δ ( ∥ ∇ l ( n , u ) ∥ L 2 2 + ∥ ( n , u ) ∥ L ∞ 2 + ∥ ( ∇ u , ∇ 2 u ) ∥ L 3 2 ) + ∥ ∇ l + 1 η ∥ L 2 2 + ∥ ( η , ∇ η ) ∥ L ∞ 2 .

Proof

We can apply the operator ∇ l to the first and second equations of equation (4.1) and multiply the first equation of (4.1) by h ′ ( ρ ¯ ) ∇ l n , and the second equation of equation (4.1) by ρ ¯ ∇ l u , to obtain

(4.6) 1 2 d d t ∫ ( h ′ ( ρ ¯ ) ∣ ∇ l n ∣ 2 + ρ ¯ ∣ ∇ l u ∣ 2 ) d x + μ ∥ ∇ l + 1 u ∥ L 2 2 + ( μ + λ ) ∥ ∇ l div u ∥ L 2 2 = ∫ ∇ l G 1 ⋅ h ′ ( ρ ¯ ) ∇ l n d x + ∫ ∇ l G 2 ⋅ ρ ¯ ∇ l u d x − ∫ ∇ l ∇ η ρ ¯ ⋅ ρ ¯ ∇ l u d x .

The term ∫ ∇ l G 1 ⋅ h ′ ( ρ ¯ ) ∇ l n d x can be rewritten as follows:

(4.7) ∫ ∇ l G 1 ⋅ h ′ ( ρ ¯ ) ∇ l n d x = − ∫ h ′ ( ρ ¯ ) ∇ l n ∇ l ( u ⋅ ∇ ( n + ρ s − ρ ¯ ) + ( n + ρ s − ρ ¯ ) div u ) d x = − ∫ h ′ ( ρ ¯ ) u ⋅ ∇ ∇ l n ∇ l n d x − ∫ h ′ ( ρ ¯ ) ∇ l n [ ∇ l , u ] ⋅ ∇ n d x − ∫ h ′ ( ρ ¯ ) ∇ l n ∇ l ( u ⋅ ∇ ( ρ s − ρ ¯ ) ) d x − ∫ h ′ ( ρ ¯ ) ∇ l n ∇ l ( ( n + ρ s − ρ ¯ ) div u ) d x

due to the commutator notation (2.2). It infers from the integration by parts that

(4.8) − ∫ h ′ ( ρ ¯ ) u ⋅ ∇ ∇ l n ∇ l n d x = − 1 2 ∫ h ′ ( ρ ¯ ) u ⋅ ∇ ∣ ∇ l n ∣ 2 d x = 1 2 ∫ h ′ ( ρ ¯ ) div u ∣ ∇ l n ∣ 2 d x ≲ ∥ div u ∥ L ∞ ∥ ∇ l n ∥ L 2 2 ≲ δ ∥ ∇ l n ∥ L 2 2 .

We employ the commutator estimates (2.3) to derive

(4.9) − ∫ h ′ ( ρ ¯ ) ∇ l n [ ∇ l , u ] ⋅ ∇ n d x ≲ ∥ ∇ l n ∥ L 2 ∥ [ ∇ l , u ] ⋅ ∇ n ∥ L 2 ≲ ∥ ∇ l n ∥ L 2 ( ∥ ∇ u ∥ L ∞ ∥ ∇ l n ∥ L 2 + ∥ ∇ l u ∥ L 6 ∥ ∇ n ∥ L 3 ) ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 ) .

Employing the product estimates (2.4) leads to

(4.10) − ∫ h ′ ( ρ ¯ ) ∇ l n ∇ l ( u ⋅ ∇ ( ρ s − ρ ¯ ) ) d x ≲ ∥ ∇ l n ∥ L 2 ∥ ∇ l ( u ⋅ ∇ ( ρ s − ρ ¯ ) ) ∥ L 2 ≲ ∥ ∇ l n ∥ L 2 ( ∥ u ∥ L ∞ ∥ ∇ l + 1 ( ρ s − ρ ¯ ) ∥ L 2 + ∥ ∇ l u ∥ L 6 ∥ ∇ ( ρ s − ρ ¯ ) ∥ L 3 ) ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ u ∥ L ∞ 2 )

and

(4.11) − ∫ h ′ ( ρ ¯ ) ∇ l n ∇ l ( ( n + ρ s − ρ ¯ ) div u ) d x ≲ ∥ ∇ l n ∥ L 2 ∥ ∇ l ( ( n + ρ s − ρ ¯ ) div u ) ∥ L 2 ≲ ∥ ∇ l n ∥ L 2 ( ∥ n + ρ s − ρ ¯ ∥ L ∞ ∥ ∇ l + 1 u ∥ L 2 + ∥ ∇ l n ∥ L 2 ∥ ∇ u ∥ L ∞ ) + ∥ ∇ l n ∥ L 2 ∥ ∇ u ∥ L 3 ∥ ∇ l ( ρ s − ρ ¯ ) ∥ L 6 ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ u ∥ L 3 2 ) .

Plugging equations (4.8)–(4.11) into equation (4.7), the term ∫ ∇ l G 1 ( h ′ ( ρ ¯ ) ∇ l n ) d x can be controlled as follows:

(4.12) ∫ ∇ l G 1 ( h ′ ( ρ ¯ ) ∇ l n ) d x ≲ δ ( ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ u ∥ L ∞ 2 + ∥ ∇ u ∥ L 3 2 ) .

It follows from the product estimates (2.4) and the interpolation inequality (2.1) of Lemma 2.1 that

(4.13) − ∫ ∇ l u ⋅ ∇ l ( u ⋅ ∇ u ) d x ≲ ∥ ∇ l u ∥ L 2 ∥ ∇ l ( u ⋅ ∇ u ) ∥ L 2 ≲ ∥ ∇ l u ∥ L 2 ( ∥ u ∥ L ∞ ∥ ∇ l + 1 u ∥ L 2 + ∥ ∇ u ∥ L 3 ∥ ∇ l u ∥ L 6 ) ≲ δ ( ∥ ∇ l u ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 )

and

(4.14) − ∫ ρ ¯ ∇ l u ⋅ ∇ l ∇ ( ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) n ) d x = ∫ ρ ¯ ∇ l + 1 u ⋅ ∇ l ( ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) n ) d x ≲ ∥ ∇ l + 1 u ∥ L 2 ∥ ∇ l ( ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) n ) ∥ L 2 ≲ ∥ ∇ l + 1 u ∥ L 2 ( ∥ h ′ ( ρ s ) − h ′ ( ρ ¯ ) ∥ L ∞ ∥ ∇ l n ∥ L 2 + ∥ ∇ l ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) ∥ L 2 ∥ n ∥ L ∞ ) ≲ δ ( ∥ n ∥ L ∞ 2 + ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 ) .

We can use the product estimates (2.4) to deduce

(4.15) − ∫ ρ ¯ ∇ l u ⋅ ∇ l ∇ ℛ d x = ∫ ρ ¯ ∇ l div u ∇ l ℛ d x ≲ ∥ ∇ l + 1 u ∥ L 2 ∥ ∇ l ℛ ∥ L 2 ≲ δ ( ∥ n ∥ L ∞ 2 + ∥ ∇ l n ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 )

by the fact (2.6) of Lemma 2.3. We can perform the similar approach to estimates (4.13)–(4.15) to conclude

(4.16) − ∫ ρ ¯ ∇ l u ⋅ ∇ l 1 n + ρ s ∇ η d x ≲ ∇ l 1 n + ρ s ∇ η L 2 ∥ ∇ l u ∥ L 2 ≲ 1 n + ρ s L ∞ ∥ ∇ l + 1 η ∥ L 2 + ∥ ∇ l ( n + ρ s ) ∥ L 2 ∥ ∇ η ∥ L ∞ ∥ ∇ l u ∥ L 2 ≲ δ ∥ ∇ l ( n , u ) ∥ L 2 2 + ∥ ∇ l + 1 η ∥ L 2 2 + ∥ ∇ η ∥ L ∞ 2

and

(4.17) ∫ ρ ¯ ∇ l u ⋅ ∇ l η ∇ Φ n + ρ s d x ≲ ∇ l η ∇ Φ n + ρ s L 2 ∥ ∇ l u ∥ L 2 ≲ ∇ Φ n + ρ s L 3 ∥ ∇ l η ∥ L 6 + ∇ l ∇ Φ n + ρ s L 2 ∥ η ∥ L ∞ ∥ ∇ l u ∥ L 2 ≲ δ ( ∥ ∇ l ( n , u ) ∥ L 2 2 + ∥ ∇ l + 1 η ∥ L 2 2 ) + ∥ η ∥ L ∞ 2 .

It follows through direct calculation that

(4.18) ∫ ρ ¯ u ⋅ 1 n + ρ s − 1 ρ ¯ ( μ Δ u + ( μ + λ ) ∇ div u ) d x ≲ ∥ u ∥ L 6 1 n + ρ s − 1 ρ ¯ L 2 ∥ ∇ 2 u ∥ L 3 ≲ δ ( ∥ ∇ 2 u ∥ L 3 2 + ∥ ∇ u ∥ L 2 2 ) .

For 1 ⩽ l ⩽ k , we employ the integration by parts and the product estimates (2.4) to obtain

∫ ρ ¯ ∇ l u ⋅ ∇ l 1 n + ρ s − 1 ρ ¯ ( μ Δ u + ( μ + λ ) ∇ div u ) d x = − ∫ ρ ¯ ∇ l + 1 u ⋅ ∇ l − 1 1 n + ρ s − 1 ρ ¯ ( μ Δ u + ( μ + λ ) ∇ div u ) d x ≲ ∥ ∇ l + 1 u ∥ L 2 ∇ l − 1 1 n + ρ s − 1 ρ ¯ ( μ Δ u + ( μ + λ ) ∇ div u ) L 2 ≲ ∥ ∇ l + 1 u ∥ L 2 1 n + ρ s − 1 ρ ¯ L ∞ ∥ ∇ l + 1 u ∥ L 2 + ∇ l − 1 1 n + ρ s − 1 ρ ¯ L 6 ∥ ∇ 2 u ∥ L 3 ≲ δ ( ∥ ∇ 2 u ∥ L 3 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ l n ∥ L 2 2 ) ,

which together with equation (4.18) yields easily that

(4.19) ∫ ρ ¯ ∇ l u ⋅ ∇ l 1 n + ρ s − 1 ρ ¯ ( μ Δ u + ( μ + λ ) ∇ div u ) d x ≲ δ ( ∥ ∇ 2 u ∥ L 3 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ l n ∥ L 2 2 )

for 0 ⩽ l ⩽ k .

As a consequence of estimates (4.13)–(4.17) and estimate (4.19), it infers that

(4.20) ∫ ∇ l G 2 ⋅ ρ ¯ ∇ l u d x ≲ δ ( ∥ ∇ l ( n , u ) ∥ L 2 2 + ∥ ∇ l + 1 u ∥ L 2 2 + ∥ ∇ u ∥ L 3 2 + ∥ n ∥ L ∞ 2 + ∥ ∇ 2 u ∥ L 3 2 ) + ∥ ∇ l + 1 η ∥ L 2 2 + ∥ η ∥ L ∞ 2 + ∥ ∇ η ∥ L ∞ 2 ,

which along with estimate (4.12) gives estimate (4.5).□

Next, under the assumptions of Theorem 1.3, we proceed to establish the dissipation estimates of n , which turn out to be indispensable in obtaining the decay rates of the solution.

Lemma 4.3

For 0 ⩽ l ⩽ k − 1 , it holds

(4.21) d d t ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∥ ∇ l + 1 n ∥ L 2 2 ≲ ∥ ∇ l + 1 ( u , η ) ∥ L 2 2 + ∥ ∇ l + 2 u ∥ L 2 2 + ∥ η ∥ L ∞ 2 + ∥ ∇ η ∥ L 3 + δ ( ∥ ∇ u ∥ L 3 2 + ∥ ∇ 2 u ∥ L 3 2 + ∥ ( n , u ) ∥ L ∞ 2 ) .

Proof

Performing the operator ∇ l to the second equation of equation (4.1), the L 2 inner product of the resulting equation with ∇ l ∇ n implies that

(4.22) ∫ ∇ l ∂ t u ⋅ ∇ l ∇ n d x + ∥ ∇ l + 1 n ∥ L 2 2 ≲ ∥ ∇ l + 2 u ∥ L 2 2 + ∥ ∇ l ( u ⋅ ∇ u ) ∥ L 2 2 + ∥ ∇ l + 1 ℛ ∥ L 2 2 + ∇ l 1 n + ρ s − 1 ρ ¯ ∇ 2 u L 2 2 + ∇ l η ∇ Φ n + ρ s L 2 2 + ∇ l 1 n + ρ s ∇ η L 2 2 + ∥ ∇ l + 1 [ ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) n ] ∥ L 2 2 .

For the first term ∫ ∇ l ∂ t u ⋅ ∇ l ∇ n d x , it is derived directly by performing the integration by parts about the time variable t and the spatial variable x

− ∫ ∇ l ∂ t u ⋅ ∇ ∇ l n d x = − d d t ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∫ ∇ l u ⋅ ∇ ∇ l ∂ t n d x = − d d t ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∫ ∇ l div u ∇ l ( u ⋅ ∇ n + n div u + ρ s div u + u ⋅ ∇ ρ s ) d x .

We can take a similar approach to equation (3.40) to conclude

(4.23) − ∫ ∇ l ∂ t u ⋅ ∇ ∇ l n d x ⩽ − d d t ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∥ ∇ l + 1 u ∥ L 2 2 + δ ( ∥ ∇ l + 1 n ∥ L 2 2 + ∥ ∇ u ∥ L 3 2 + ∥ u ∥ L ∞ 2 ) .

By the product estimates (2.4), one can arrive at

(4.24) ∇ l 1 ρ ¯ − 1 n + ρ s Δ u L 2 ≲ 1 ρ ¯ − 1 n + ρ s L ∞ ∥ ∇ l + 2 u ∥ L 2 + ∇ l 1 ρ ¯ − 1 n + ρ s L 6 ∥ ∇ 2 u ∥ L 3 ≲ δ ( ∥ ∇ l + 2 u ∥ L 2 + ∥ ∇ l + 1 n ∥ L 2 + ∥ ∇ 2 u ∥ L 3 ) ,

and similarly,

(4.25) ∥ ∇ l + 1 [ ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) n ] ∥ L 2 ≲ ∥ h ′ ( ρ s ) − h ′ ( ρ ¯ ) ∥ L ∞ ∥ ∇ l + 1 n ∥ L 2 + ∥ ∇ l + 1 ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) ∥ L 2 ∥ n ∥ L ∞ ≲ δ ( ∥ n ∥ L ∞ + ∥ ∇ l + 1 n ∥ L 2 ) .

We can deal with the remaining nonlinear terms like equations (3.43) and (3.44), which combined with estimates (4.23)–(4.25) prove estimate (4.21).□

Finally, using the similar approach to equation (3.34) in Lemma 3.2, the energy estimates of η can be established in the following lemma.

Lemma 4.4

Under the assumption of Theorem 1.3, for 0 ⩽ l ⩽ k , it holds that

(4.26) d d t ∫ ∣ ∇ l η ∣ 2 d x + ∥ ∇ l + 1 η ∥ L 2 2 ≲ ∥ η ∥ L ∞ 2 + δ ∥ ∇ l + 1 u ∥ L 2 2 .

Proof

Applying the operator ∇ l to the third equation of (4.1) and multiplying ∇ l η to the resulting identities, we obtain

1 2 d d t ∫ ∣ ∇ l η ∣ 2 d x + ∥ ∇ l + 1 η ∥ L 2 2 = − ∫ ∇ l ( η ∇ Φ ) ⋅ ∇ l + 1 η d x + ∫ ∇ l ( η u ) ⋅ ∇ l + 1 η d x ≲ ∥ ∇ l ( η ∇ Φ ) ∥ L 2 ∥ ∇ l + 1 η ∥ L 2 + ∥ ∇ l ( η u ) ∥ L 2 ∥ ∇ l + 1 η ∥ L 2 ≲ ( ∥ ∇ l η ∥ L 6 ∥ ∇ Φ ∥ L 3 + ∥ η ∥ L ∞ ∥ ∇ l ∇ Φ ∥ L 2 ) ∥ ∇ l + 1 η ∥ L 2 + δ ∥ ∇ l + 1 ( u , η ) ∥ L 2 2 ≲ ∥ η ∥ L ∞ 2 + δ ∥ ∇ l + 1 ( u , η ) ∥ L 2 2 .

To this end, estimate (4.26) is proved.□

From Lemmas 4.2–4.4, we derive the following proposition.

Proposition 4.5

There exists an energy functional ℰ ℓ k equivalent to ∥ ∇ ℓ ( n , u , η ) ∥ H k − ℓ 2 such that

(4.27) d d t ℰ ℓ k + ∥ ∇ ℓ + 1 n ∥ H k − ℓ − 1 2 + ∥ ∇ ℓ + 1 ( u , η ) ∥ H k − ℓ 2 ≲ δ ( ∥ ∇ ℓ ( n , u ) ∥ L 2 2 + ∥ ( n , u ) ∥ L ∞ 2 + ∥ ∇ u ∥ L 3 2 + ∥ ∇ 2 u ∥ L 3 2 ) + ∥ ( η , ∇ η ) ∥ L ∞ 2 + ∥ ∇ η ∥ L 3 2 .

Proof

Taking l from ℓ to k , summing up estimate (4.5) of Lemma 4.2 and estimate (4.26) of Lemma 4.4, it shows that

(4.28) d d t ( h ′ ( ρ ¯ ) ∥ ∇ ℓ n ∥ H k − ℓ 2 + ρ ¯ ∥ ∇ ℓ u ∥ H k − ℓ 2 + ∥ ∇ ℓ η ∥ H k − ℓ 2 ) + ∥ ∇ ℓ + 1 ( u , η ) ∥ H k − ℓ 2 ≲ δ ( ∥ ∇ ℓ ( n , u ) ∥ L 2 2 + ∥ ( n , u ) ∥ L ∞ 2 + ∥ ∇ u ∥ L 3 2 + ∥ ∇ 2 u ∥ L 3 2 ) + ∥ η ∥ L ∞ 2 + ∥ ∇ η ∥ L ∞ 2 .

It follows from estimate (4.21) of Lemma 4.3 that

(4.29) d d t ∑ ℓ ⩽ l ⩽ k − 1 ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∑ ℓ + 1 ⩽ l ⩽ k ∥ ∇ l n ∥ L 2 2 ≲ ∑ ℓ + 1 ⩽ l ⩽ k + 1 ∥ ∇ ℓ u ∥ L 2 2 + ∑ ℓ + 1 ⩽ l ⩽ k ∥ ∇ ℓ η ∥ L 2 2 + ∥ η ∥ L ∞ 2 + ∥ ∇ η ∥ L 3 2 + δ ( ∥ ( n , u ) ∥ L ∞ 2 + ∥ ∇ u ∥ L 3 2 + ∥ ∇ 2 u ∥ L 3 2 ) ,

provided the index l is taken from ℓ to k − 1 .

Combining the inequalities (4.29) and (4.28), since δ is small, we deduce that

(4.30) d d t ( h ′ ( ρ ¯ ) ∥ ∇ ℓ n ∥ H k − ℓ 2 + ρ ¯ ∥ ∇ ℓ u ∥ H k − ℓ 2 + ∥ ∇ ℓ η ∥ H k − ℓ 2 ) + ε ∑ ℓ ⩽ l ⩽ k − 1 ∫ ∇ l u ⋅ ∇ ∇ l n d x + ∥ ∇ ℓ + 1 n ∥ H k − ℓ − 1 2 + ∥ ∇ ℓ + 1 ( u , η ) ∥ H k − ℓ 2 ≲ δ ( ∥ ∇ ℓ ( n , u ) ∥ L 2 2 + ∥ ( n , u ) ∥ L ∞ 2 + ∥ ∇ u ∥ L 3 2 + ∥ ∇ 2 u ∥ L 3 2 ) + ∥ ( η , ∇ η ) ∥ L ∞ 2 + ∥ ∇ η ∥ L 3 2 .

We define ℰ ℓ k to be the expression under the time derivative in equation (4.30). Since ε is small, ℰ ℓ k is equivalent to ∥ ∇ ℓ ( n , u , η ) ∥ H ℓ − k 2 , then we complete the proof of Proposition 4.5.□

In order to obtain the decay rates of the solution ( n , u , η ) T , we should give the estimates of the nonlinear terms ∥ ( G 1 , G 2 , F 3 ) ∥ L r and ∥ ∇ ℓ ( G 1 , G 2 , F 3 ) ∥ L 2 in the expression (4.3) of Proposition 4.1.

Lemma 4.6

It holds that for 1 ⩽ r < 6 ∕ 5 ,

(4.31) ∥ ( G 1 , G 2 , F 3 ) ∥ L r ≲ δ ( ∥ ( n , u , η ) ∥ L ∞ + ∥ ∇ η ∥ L ∞ ) + δ ∇ 4 − 3 r ( n , u , η ) L 2 + ∇ 5 − 3 r u L 2 .

For 0 ⩽ ℓ ⩽ 1 , we have

(4.32) ∥ ∇ ℓ ( G 1 , G 2 , F 3 ) ∥ L 2 ≲ δ ( ∥ ∇ ℓ + 1 ( n , u , η ) ∥ L 2 + ∥ ∇ ℓ + 2 u ∥ L 2 + ∥ ( n , u , η ) ∥ L ∞ ) + δ ( ∥ ∇ η ∥ L 3 + ∥ ∇ 2 u ∥ L 3 ) .

Proof

Under the conditions of Theorem 1.3, we can use the Hölder’s inequality and the interpolation inequality (2.1) to deduce

(4.33) ∥ ( ρ s − ρ ¯ ) div u ∥ L r + ∥ ∇ ( ρ s − ρ ¯ ) u ∥ L r ≲ ∥ ρ s − ρ ¯ ∥ L 2 ∥ ∇ u ∥ L 1 1 ∕ r − 1 ∕ 2 + ∥ ∇ ( ρ s − ρ ¯ ) ∥ L r ∥ u ∥ L ∞ ≲ δ ∇ 4 − 3 r u L 2 + ∥ u ∥ L ∞

and

(4.34) 1 n + ρ s − 1 ρ ¯ ∇ 2 u L r ≲ 1 n + ρ s − 1 ρ ¯ L 2 ∥ ∇ 2 u ∥ L 1 1 ∕ r − 1 ∕ 2 ≲ δ ∇ 5 − 3 r u L 2 .

Similar to equations (4.33) and (4.34), it is straightforward to obtain

(4.35) ∥ ∇ ( ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) n ) ∥ L r ≲ ∥ h ′ ( ρ s ) − h ′ ( ρ ¯ ) ∥ L 2 ∥ ∇ n ∥ L 1 1 ∕ r − 1 ∕ 2 + ∥ ∇ ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) ∥ L r ∥ n ∥ L ∞ ≲ δ ∇ 4 − 3 r n L 2 + ∥ n ∥ L ∞

and

(4.36) ∥ div ( η ∇ Φ ) ∥ L r ≲ ∥ ∇ 2 Φ ∥ L r ∥ η ∥ L ∞ + ∥ ∇ Φ ∥ L r ∥ ∇ η ∥ L ∞ ≲ δ ( ∥ η ∥ L ∞ + ∥ ∇ η ∥ L ∞ ) .

In the same manner, the remaining nonlinear terms can also be estimated and estimate (4.31) will be obtained.

For 0 ⩽ ℓ ⩽ 1 , we can employ the product estimate (2.4) to obtain

(4.37) ∥ ∇ ℓ div ( ( n + ρ s − ρ ¯ ) u ) ∥ L 2 ≲ ∥ ( n + ρ s − ρ ¯ ) ∥ L 3 ∥ ∇ ℓ + 1 u ∥ L 6 + ∥ ∇ ℓ + 1 ( n + ρ s − ρ ¯ ) ∥ L 2 ∥ u ∥ L ∞ ≲ δ ( ∥ ∇ ℓ + 2 u ∥ L 2 + ∥ ∇ ℓ + 1 n ∥ L 2 + ∥ u ∥ L ∞ )

and

∥ ∇ ℓ ∇ ( ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) n ) ∥ L 2 ≲ ∥ ∇ ℓ + 1 ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) ∥ L 2 ∥ n ∥ L ∞ + ∥ ( h ′ ( ρ s ) − h ′ ( ρ ¯ ) ) ∥ L ∞ ∥ ∇ ℓ + 1 n ∥ L 2 ≲ δ ( ∥ n ∥ L ∞ + ∥ ∇ ℓ + 1 n ∥ L 2 ) .

Adopting a similar approach, it can be obtained by direct calculation that

∇ ℓ 1 n + ρ s − 1 ρ ¯ ∇ 2 u L 2 ≲ ∇ ℓ 1 n + ρ s − 1 ρ ¯ L 6 ∥ ∇ 2 u ∥ L 3 + 1 n + ρ s − 1 ρ ¯ L ∞ ∥ ∇ ℓ + 2 u ∥ L 2 ≲ δ ( ∥ ∇ 2 u ∥ L 3 + ∥ ∇ ℓ + 1 n ∥ L 2 + ∥ ∇ ℓ + 2 u ∥ L 2 )

and

(4.38) ∇ ℓ η ∇ Φ n + ρ s L 2 ≲ ∇ ℓ ∇ Φ n + ρ s L 2 ∥ η ∥ L ∞ + ∇ Φ n + ρ s L 3 ∥ ∇ ℓ η ∥ L 6 ≲ ( ∥ ∇ ℓ ∇ Φ ∥ L 2 ∥ n + ρ s ∥ L ∞ + ∥ ∇ ℓ ( n , ρ s ) ∥ L 6 ∥ ∇ Φ ∥ L 3 ) ∥ η ∥ L ∞ + ∥ ∇ Φ ∥ L 3 ∥ ∇ ℓ η ∥ L 6 ≲ δ ( ∥ ∇ ℓ + 1 ( n , η ) ∥ L 2 + ∥ η ∥ L ∞ ) .

The remaining terms can be obtained in a similar way and estimate (4.32) is proved.□

4.2 Proof of Theorem 1.3

In this subsection, we will prove Theorem 1.3. Let k ⩾ 3 and 0 ⩽ ℓ ⩽ 1 . Adding ∥ ∇ ℓ ( n , u , η ) ∥ L 2 2 to both sides of estimate (4.27) of Proposition 4.5, there exists some constant λ > 0 such that

(4.39) d d t ℰ ℓ k + λ ℰ ℓ k ≲ ∥ ∇ ℓ ( n , u , η ) ∥ L 2 2 + ∥ ( n , u , η ) ∥ L ∞ 2 + ∥ ∇ ( u , η ) ∥ L 3 2 + ∥ ∇ 2 u ∥ L 3 2 + ∥ ∇ η ∥ L ∞ 2 .

By the Gronwall inequality, it gives

(4.40) ∥ ∇ ℓ ( n , u , η ) ∥ H k − ℓ 2 ≲ e − λ t ∥ ∇ ℓ ( n 0 , u 0 , η 0 ) ∥ H k − ℓ 2 + ∫ 0 t e − λ ( t − τ ) ( ∥ ∇ ℓ ( n , u , η ) ∥ L 2 2 + ∥ ( n , u , η , ∇ η ) ∥ L ∞ 2 + ∥ ( ∇ u , ∇ η , ∇ 2 u ) ∥ L 3 2 ) d τ .

For simplicity of notation, we denote

(4.41) ℒ ( t ) ≔ ∥ ∇ ( n , u , η ) ∥ H k − 1

and define

(4.42) ℳ ( t ) ≔ ∥ ∇ ( n , u , η ) ∥ L 2 + ∥ ( n , u , η ) ∥ L ∞ + ∥ ∇ ( u , η ) ∥ L 3 + ∥ ∇ 2 u ∥ L 3 + ∥ ∇ η ∥ L ∞ .

The expression N is defined as follows:

(4.43) N ( t ) ≔ sup 0 ⩽ τ ⩽ t ( ( 1 + τ ) ζ + 1 ∕ 2 ( ℒ + ℳ ) + ( 1 + τ ) ζ ∥ ( n , u , η ) ∥ L 2 ) ,

where

(4.44) ζ = 3 2 1 max ( p , r ) − 1 2 .

In terms of equations (4.41)–(4.42), it follows from equation (4.40) with ℓ = 1 that

(4.45) ℒ 2 ( t ) ≲ e − λ t K 0 2 + ∫ 0 t e − λ ( t − τ ) ℳ 2 ( τ ) d τ ,

where the notation

(4.46) K 0 = ∥ ( n 0 , u 0 , η 0 ) ∥ L p + ∥ ( n 0 , u 0 , η 0 ) ∥ H k .

From the decay property in Proposition 4.1, together with estimates (4.31) and (4.32) of the nonlinear terms, we can obtain

(4.47) ∥ ( n , u , η ) ( t ) ∥ L 2 ≲ ( 1 + t ) − 3 2 1 p − 1 2 K 0 + ∫ 0 t ( 1 + t − τ ) − 3 2 1 r − 1 2 δ ( ℳ + ℒ ) ( τ ) d τ ≲ ( 1 + t ) − 3 2 1 p − 1 2 K 0 + δ ∫ 0 t ( 1 + t − τ ) − 3 2 1 r − 1 2 ( 1 + τ ) − ζ − 1 2 N ( τ ) d τ ≲ ( 1 + τ ) − ζ ( K 0 + δ N ( t ) ) .

Taking ℓ = 1 in Proposition 4.1, in terms of estimates (4.31) and (4.32) and the definition (4.43), we can obtain the decay estimates of ∥ ∇ ( n , u , η ) ( t ) ∥ L 2

(4.48) ∥ ∇ ( n , u , η ) ( t ) ∥ L 2 ≲ ( 1 + t ) − 3 2 1 p − 1 2 − 1 2 K 0 + ∫ 0 t ( 1 + t − τ ) − 3 2 1 r − 1 2 − 1 2 δ ( ℒ + ℳ ) ( τ ) d τ ≲ ( 1 + t ) − 3 2 1 p − 1 2 − 1 2 K 0 + δ ∫ 0 t ( 1 + t − τ ) − 3 2 1 r − 1 2 − 1 2 ( 1 + τ ) − ζ − 1 2 N ( t ) d τ ≲ ( 1 + t ) − ζ − 1 2 ( K 0 + δ N ( t ) ) .

We can take a similar approach to equation (4.48) to deduce

(4.49) ∥ ( n , u , η ) ∥ L ∞ + ∥ ∇ ( u , η ) ∥ L 3 + ∥ ∇ 2 u ∥ L 3 + ∥ ∇ η ∥ L ∞ ≲ ∥ ∇ 3 ∕ 2 ( n , u , η ) ∥ L 2 + ∥ ∇ 5 ∕ 2 ( u , η ) ∥ L 2 ≲ ( 1 + t ) − ζ − 3 4 K 0 + ∫ 0 t ( 1 + t − τ ) − 3 2 1 r − 1 2 − 3 4 δ ( ℒ + ℳ ) ( τ ) d τ ≲ ( 1 + t ) − ζ − 3 4 K 0 + δ ∫ 0 t ( 1 + t − τ ) − 3 2 1 r − 1 2 − 3 4 ( 1 + τ ) − ζ − 1 2 N ( t ) d τ ≲ ( 1 + t ) − ζ − 1 2 ( K 0 + δ N ( t ) ) .

In terms of the decay estimates (4.47)–(4.49), it is evident that

(4.50) ℳ ( t ) ≲ ( 1 + t ) − ζ − 1 2 ( K 0 + δ N ( t ) ) ,

which together with equation (4.45) gives

(4.51) ℒ 2 ≲ e − λ t K 0 2 + ∫ 0 t e − λ ( t − τ ) ( 1 + τ ) − 2 ( ζ + 1 ∕ 2 ) ( K 0 2 + δ N 2 ( t ) ) d τ ≲ ( 1 + t ) − 2 ( ζ + 1 ∕ 2 ) ( K 0 2 + δ N 2 ( t ) ) .

Therefore, by the definition of N , in view of equations (4.47), (4.50), and (4.51), we deduce

N ( t ) ≲ K 0 + δ N ( t ) ,

which yields

N ≲ K 0 ,

since δ is sufficiently small. To this end, we have completed the proof of Theorem 1.3.

Funding information: The author was supported by the National Natural Science Foundation of China (Grant No. 12001077) and the Natural Science Foundation of Chongqing (Grant No. CSTB2023NSCQ-MSX0575).
Conflict of interest: The author states that there is no conflict of interest.

References

[1] M. Abdelwahed and N. Chorfi, Spectral discretization of the time-dependent Navier-Stokes problem with mixed boundary conditions, Adv. Nonlinear Anal. 11 (2022), no. 1, 1447–1465. 10.1515/anona-2022-0253Search in Google Scholar

[2] J. Ballew, Mathematical Topics in Fluid-Particle Interaction, Thesis (Ph.D.)–University of Maryland, College Park, 2014, 130pp. Search in Google Scholar

[3] J. Ballew and K. Trivisa, Weakly dissipative solutions and weak-strong uniqueness for the Navier Stokes-Smoluchowski system, Nonlinear Anal. 91 (2013), 1–19. 10.1016/j.na.2013.06.002Search in Google Scholar

[4] S. Berres, R. Bürger, K. H. Karlsen, and E. M. Rory, Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression, SIAM J. Appl. Math. 64 (2003), no. 1, 41–80. 10.1137/S0036139902408163Search in Google Scholar

[5] J. A. Carrillo and T. Goudon, Stability and asymptotic analysis of a fluid–particle interaction model, Comm. Partial Differential Equations 31 (2006), no. 7–9, 1349–1379. 10.1080/03605300500394389Search in Google Scholar

[6] J. A. Carrillo, T. Karper, and K. Trivisa, On the dynamics of a fluid–particle interaction model: The bubbling regime, Nonlinear Anal. 74 (2011), no. 8, 2778–2801. 10.1016/j.na.2010.12.031Search in Google Scholar

[7] Y. S. Chen, S. J. Ding, and W. J. Wang, Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307. 10.3934/dcds.2016032Search in Google Scholar

[8] M. Chen, Z. L. Liang, D. H. Wang, and R. Z. Xu, Energy equality in compressible fluids with physical boundaries, SIAM J. Math. Anal. 52 (2020), no. 2, 1363–1385. 10.1137/19M1287213Search in Google Scholar

[9] H. J. Choe and H. Kim, Strong solutions of the Navier-Stokes equations for isentropic compressible fluids, J. Differential Equations 190 (2003), no. 2, 504–523. 10.1016/S0022-0396(03)00015-9Search in Google Scholar

[10] P. Constantin and N. Masmoudi, Global well-posedness for a Smoluchowski equation coupled with Navier-Stokes equations in 2D, Comm. Math. Phys. 278 (2008), no. 1, 179–191. 10.1007/s00220-007-0384-2Search in Google Scholar

[11] K. Deckelnick, L2-decay for the compressible Navier-Stokes equations in unbounded domains, Comm. Partial Differential Equations, 18 (1993), no. 9–10, 1445–1476. 10.1080/03605309308820981Search in Google Scholar

[12] S. J. Ding, B. Y. Huang, and Q. R. Li, Global existence and decay estimates for the classical solutions to a compressible fluid–particle interaction model, Acta Math. Sci. Ser. B (Engl. Ed.) 39 (2019), no. 6, 1525–1537. 10.1007/s10473-019-0605-8Search in Google Scholar

[13] S. J. Ding, B. Y. Huang, and H. Y. Wen, Global well-posedness of classical solutions to a fluid–particle interaction model in R3, J. Differential Equations 263 (2017), no. 12, 8666–8717. 10.1016/j.jde.2017.08.048Search in Google Scholar

[14] W. C. Dong and Z. H. Guo, Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data, Adv. Nonlinear Anal. 12 (2023), no. 1, 132–168. 10.1515/anona-2022-0246Search in Google Scholar

[15] S. Doboszczak, Existence and Weak-Strong Uniqueness for the Navier-Stokes-Smoluchowski System Over Moving Domains, Thesis (Ph.D.)–University of Maryland, College Park, 2016, 109pp. Search in Google Scholar

[16] R. J. Duan, H. X. Liu, S. Ukai, and T. Yang, Optimal Lp-Lq convergence rates for the compressible Navier-Stokes equations with potential force, J. Differential Equations 238 (2007), no. 1, 220–233. 10.1016/j.jde.2007.03.008Search in Google Scholar

[17] R. J. Duan, S. Ukai, T. Yang, and H. J. Zhao, Optimal convergence rates for the compressible Navier-Stokes equations with potential force, Math. Models Methods Appl. Sci. 17 (2007), no. 5, 737–758. 10.1142/S021820250700208XSearch in Google Scholar

[18] D. Y. Fang, R. Z. Zi, and T. Zhang, Global classical large solutions to a 1D fluid–particle interaction model: The bubbling regime, J. Math. Phys. 53 (2012), no. 3, 033706, 21pp. 10.1063/1.3693979Search in Google Scholar

[19] E. Feireisl, A. Novotnć and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech. 3 (2001), no. 4, 358–392. 10.1007/PL00000976Search in Google Scholar

[20] A. L. Fogelson, Continuum models of platelet aggregation: formulation and mechanical properties, SIAM J. Appl. Math. 52 (1992), no. 4, 1089–1110. 10.1137/0152064Search in Google Scholar

[21] Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Comm. Partial Differential Equations 37 (2012), no. 12, 2165–2208. 10.1080/03605302.2012.696296Search in Google Scholar

[22] D. Hoff, Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data, J. Differential Equations, 120 (1995), no. 1, 215–254. 10.1006/jdeq.1995.1111Search in Google Scholar

[23] D. Hoff and K. Zumbrun, Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J. 44 (1995), no. 2, 603–676. 10.1512/iumj.1995.44.2003Search in Google Scholar

[24] B. Y. Huang, S. J. Ding, and H. Y. Wen, Local classical solutions of compressible Navier-Stokes-Smoluchowski equations with vacuum, Discrete Contin. Dyn. Syst. Ser. S 9 (2016), no. 6, 1717–1752. 10.3934/dcdss.2016072Search in Google Scholar

[25] X. D. Huang and J. Li, Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, Arch. Ration. Mech. Anal. 227 (2018), no. 3, 995–1059. 10.1007/s00205-017-1188-ySearch in Google Scholar

[26] X. D. Huang, J. Li and Z. P. Xin, Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math. 65 (2012), no. 4, 549–585. 10.1002/cpa.21382Search in Google Scholar

[27] B. K. Huang, L. Q. Liu, and L. Zhang, On the existence of global strong solutions to 2D compressible Navier-Stokes-Smoluchowski equations with large initial data, Nonlinear Anal. Real World Appl. 49 (2019), 169–195. 10.1016/j.nonrwa.2019.03.005Search in Google Scholar

[28] P. E. Jabin, Various levels of models for aerosols, Math. Models Methods Appl. Sci. 12 (2002), no. 7, 903–919. 10.1142/S0218202502001957Search in Google Scholar

[29] N. Ju, Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Comm. Math. Phys. 251 (2004), no. 2, 365–376. 10.1007/s00220-004-1062-2Search in Google Scholar

[30] S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, Kyoto University, Doctoral Thesis, 1984. Search in Google Scholar

[31] T. Kobayashi and Y. Shibata, Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R3, Comm. Math. Phys. 200 (1999), no. 3, 621–659. 10.1007/s002200050543Search in Google Scholar

[32] J. K. Li, Global small solutions of heat conductive compressible Navier-Stokes equations with vacuum: Smallness on scaling invariant quantity, Arch. Ration. Mech. Anal. 237 (2020), no. 2, 899–919. 10.1007/s00205-020-01521-7Search in Google Scholar

[33] J. Li and Z. L. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl. 102 (2014), no. 4, 640–671. 10.1016/j.matpur.2014.02.001Search in Google Scholar

[34] J. Li and A. Matsumura, On the Navier-Stokes equations for three-dimensional compressible barotropic flow subject to large external potential forces with discontinuous initial data, J. Math. Pures Appl. 95 (2011), no. 5, 495–512. 10.1016/j.matpur.2010.12.002Search in Google Scholar

[35] J. Li and Z. P. Xin, Global well-posedness and large time asymptotic behavior of classical solutions to the compressible Navier-Stokes equations with vacuum, Ann. PDE 5 (2019), no. 1, Paper no. 37, 37 pp. 10.1007/s40818-019-0064-5Search in Google Scholar

[36] Y. Liu, Local well-posedness to the Cauchy problem of the 2D compressible Navier-Stokes–Smoluchowski equations with vacuum, J. Math. Anal. Appl. 489 (2020), Paper no. 124154, 24 pp. 10.1016/j.jmaa.2020.124154Search in Google Scholar

[37] T. P. Liu and W. K. Wang, The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions, Comm. Math. Phys. 196 (1998), no. 1, 145–173. 10.1007/s002200050418Search in Google Scholar

[38] Y. Liu, L. Zhang, X. Wang, and W. K. Liu, Coupling of Navier-Stokes equations with protein molecular dynamics and its application to hemodynamics, J. Numer. Meth. Fluids 46 (2004), 1237–1252. 10.1002/fld.798Search in Google Scholar

[39] S. Y. Ma, J. W. Sun, and H. M. Yu, Global existence and stability of temporal periodic solution to non-isentropic compressible Euler equations with a source term, Commun. Anal. Mech. 15 (2023), no. 2, 245–266. 10.3934/cam.2023013Search in Google Scholar

[40] A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat conductive gases, J. Math. Kyoto Univ. 20 (1980), no. 1, 67–104. 10.1215/kjm/1250522322Search in Google Scholar

[41] A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys. 89 (1983), no. 4, 445–464. 10.1007/BF01214738Search in Google Scholar

[42] L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115–162. Search in Google Scholar

[43] P. J. O’Rourke, Collective Drop Effects on Vaporizing Liquid Sprays, Dissertation, Princeton University, 1981. Search in Google Scholar

[44] G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal. 9 (1985), no.5, 399–418. 10.1016/0362-546X(85)90001-XSearch in Google Scholar

[45] R. Salvi and I. Straškraba, Global existence for viscous compressible fluids and their behavior as t→∞, IA Math. 40 (1993), no. 1, 17–51. Search in Google Scholar

[46] J. Serrin, On the uniqueness of compressible fluid motion, Arch. Rational Mech. Anal. 3 (1959), 271–288. 10.1007/BF00284180Search in Google Scholar

[47] Z. Tan, Y. Wang, and L. L. Tong, Decay estimates of solutions to the bipolar non-isentropic compressible Euler-Maxwell system, Nonlinearity 30 (2017), no. 10, 3743–3772. 10.1088/1361-6544/aa7effSearch in Google Scholar

[48] Z. Tan, Y. J. Wang, and Y. Wang, Stability of steady states of the Navier-Stokes-Poissonequations with non-flat doping profile, SIAM J. Math. Anal. 47 (2015), no. 1, 179–209. 10.1137/130950069Search in Google Scholar

[49] K. Watanabe, Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension, Adv. Nonlinear Anal. 12 (2023), no. 1, Paper no. 20220279, 35. 10.1515/anona-2022-0279Search in Google Scholar

[50] X. H. Yang, Local well-posedness of the compressible Navier-Stokes-Smoluchowski equations with vacuum, J. Math. Anal. Appl. 485 (2020), no. 1, Paper no. 123792, 8. 10.1016/j.jmaa.2019.123792Search in Google Scholar

Received: 2022-09-26

Revised: 2023-10-18

Accepted: 2024-01-03

Published Online: 2024-02-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3

Abstract

1 Introduction

Theorem 1.1

Remark 1.2

Theorem 1.3

Remark 1.4

2 Preliminaries

Lemma 2.1

Proof

Lemma 2.2

Proof

Lemma 2.3

Proof

Lemma 2.4

3 Global existence of the classical solution

3.1 Energy estimates

Lemma 3.1

Proof

Lemma 3.2

Proof

Lemma 3.3

Proof

Lemma 3.4

Proof

3.2 Proof of Theorem 1.1

4 Time decay rates of the solution

Proposition 4.1

4.1 Energy estimates

Lemma 4.2

Proof

Lemma 4.3

Proof

Lemma 4.4

Proof

Proposition 4.5

Proof

Lemma 4.6

Proof

4.2 Proof of Theorem 1.3

References

Journal and Issue

Articles in the same Issue

Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ³