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

: The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of ﬂ uid 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 x , 0, 0 s T ( ( ) ) is in a small neighborhood of the constant state ρ ¯, 0, 0 T ( ) at in ﬁ nity, 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.


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]: 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 = x Φ Φ( ) 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 x Φ( ) affects the particles and the fluid.The viscosity coefficients μ and λ satisfy > + ⩾ μ μ λ 0, 2 3 0.
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 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 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 D 3 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 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 2 and by Yang [50] in D 2 bounded domain; furthermore, the global existence of the strong solution with large initial data, which may contain vacuum in 2 , was obtained in [27].Fang et al. [18] investigated the global classical large solution with initial density containing vacuum in D 1 space.Huang et al. [24] studied the Cauchy problem of equations (1.1) with vacuum in 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 3 was established in [13] when the initial data were of small energy around the steady state ρ x η x , 0, s s T ( ( ) ( )) .Chen et al. [7] obtained the global classical solution for the Cauchy problem of equation (1.1) with the stationary solution ρ x , 0, 0 s 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 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 3 and looking for the global classical solution near the stationary solution ρ x , 0, 0 s 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 , , ¯, 0, 0 as for the positive constant ρ ¯.Similar to the statement in [7,41], there exists the steady state solution ρ , 0, 0 and the stationary solution ρ , 0, 0 We define the perturbation by then problem (1.1) can be reformulated into the perturbed form of where By Taylor expansion, we have Notation.We use the notation ≲ x y to express ⩽ x Cy with some positive constant > C 0. The norm x y , ∥ ∥ ∥ ∥ Our first result in this article is concerned with the global existence of the classical solution near the steady state ρ , 0, 0 s T ( ) .
Theorem 1.1.Under the condition that , 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, 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 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 ⩽ < ∕ p 1 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 Under the assumptions of Theorem 1.1, if further assumptions are imposed on Φ and the initial data n u , 0 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 6 5 is sufficiently small and n u , 1 .
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 ρ s 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.

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.
with the superscript ⩽ ⩽ ϑ 0 1 and the constant α satisfies Proof.The proof of this lemma can be seen in the study by Nirenberg [42].□ Lemma 2.2.The commutator notation is defined as for the integer ⩾ l 1.Then, the commutator estimates and the product estimates hold.In the above, ∈ +∞ p p p , , )such that Proof.See Lemma 3.1 in the study by Ju [29].□ Lemma 2.3.We have that for ⩾ l 1, and 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: ) be a smooth function of n θ , with bounded derivatives of any order, then for any integer ⩾ k 1 and ⩽ ⩽ ∞ p 2 , we have

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.

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 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 and for = l k 1, 2,…, , one can obtain Decay estimates of Navier-Stokes-Smoluchowski equations  5 Proof.Applying the operator ∇ l with ⩽ ⩽ l k 0 to first and second equation in equation (1.7), and taking the inner product with ′ ∇ h ρ n s l ( ) and ∇ ρ u s l , respectively, we can obtain the following identity directly: If = l 0, by cancellation of the second term on the left-hand side of the identity (3.4), it shows that By Hölder's inequality and the interpolation inequality, we can estimate The following inequality is a simple consequence of the Hölder's inequality and Young's inequality It follows from the integration by parts and the Hölder's inequality that Therefore, the inequality (3.2) follows from estimates (3.6)- (3.8).
It remains to deal with the case ⩽ ⩽ l k 1 .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 We obtain after the Hölder's inequality and the commutator estimates (2.3) that for = l 1, 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 and .
In terms of estimates (3.9)- (3.11), it implies that We can employ the integration by parts and the commutator notation (2.2) to rewrite and 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 ⩽ ⩽ l k 2 : Combining the product estimates (2.4) and the Gagliardo-Nirenberg's inequality (2.1), due to the Young's inequality, for In terms of estimates (3.13)-(3.17), it implies that and, similarly, From the product estimates (2.4), the Gagliardo-Nirenberg's inequality (2.1), and the Young's inequality, one can conclude that Now, we turn to estimate the term We use the commutator notation (2.2) to rewrite It infers from the integration by parts and the Hölder's inequality that By the application of the interpolation inequality (2.1) and the commutator estimates (2.3), we can deduce that Decay estimates of Navier-Stokes-Smoluchowski equations  9 It implies from the product estimates (2.4) that It is derived from the product estimates (2.4) along with the Young's inequality that We can deduce from equation (2.5) of Lemma 2.3 that We should distinguish the order l while estimating the following term: , by employing the integration by parts and the product estimates (2.4), it can be estimated as follows: From estimates (3.27) and (3.28), we have Similarly to equation (3.29), it holds One can deduce from the Hölder's inequality and the product estimates (2.4) that and where the Young's inequality is also be used.To this end, in light of estimate (3.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
Proof.Taking the inner product of the third equation in equation (1.7) with η, together with the integration by parts, we obtain which implies equation (3.33), since δ and ε are sufficiently small.The energy estimate (3.34) at l level with ⩽ ⩽ l k 1 can be obtained as follows: The energy estimates established in Lemmas 3.1-3.2are 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 Proof.Taking the inner product of the second equation in equation , , .
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 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 ⩽ ⩽ − l k 1 1can also be obtained.
Lemma 3.4.For integer l with ⩽ ⩽ − l k 1 1 , under the assumption of equation (3.1), we conclude Proof.Applying ∇ l with ⩽ ⩽ − l k 1 1to (1.7) 2 and then taking the L 2 inner product with ∇∇ n l , it gives We can take a similar approach to equation (3.37), using the Hölder's inequality and the product estimates (2.4), to obtain By the commutator estimates (2.3), it infers With the help of the product estimates (2.4) and the Young's inequality, we deduce and Decay estimates of Navier-Stokes-Smoluchowski equations  13 We can adopt a similar approach to equations (3.42) and (3.43) to have

Proof of Theorem 1.1
The summation of equations (3.2) and ( 3. A suitable linear combination of equations (3.45) and (3.46) implies that which combined with equation (3.47) gives In order to obtain the uniform estimates of the solution n u η , , T ( ) , we assume ) ) ) ) which along with definition (3.49) gives Since δ is sufficiently small, the term δZ t ) can be absorbed, and we directly deduce With the help of equation (3.50) and taking the integration of equation (3.48) about the time variable t, we have To this end, we have obtained the uniform estimates of the solution and finished proof of Theorem 1.1.

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: , , , with the nonlinear terms G 1 and G 2 denoted by By the Duhamel principle, the solution n u η , , T ( ) to equation (4.1) can be expressed as follows: Decay estimates of Navier-Stokes-Smoluchowski equations  15 where is defined by 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.

Energy estimates
We should make a little preparations for the later use in the proof of Theorem 1. .
Under the further assumption that 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 ⩽ ⩽ l k 0 , and under the assumption of Theorem 1.3, we have 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 n ¯l ( ) , and the second equation of equation (4.1) by ∇ ρ u ¯l , to obtain The term ∫∇ ⋅ ′ ∇ G h ρ n x ¯d l l 1 ( ) can be rewritten as follows: due to the commutator notation (2.2).It infers from the integration by parts that We employ the commutator estimates (2.3) to derive Employing the product estimates (2.4) leads to Plugging equations (4.8)-(4.11)into equation (4.7), the term ∫∇ ′ ∇ G h ρ n x ¯d l l 1 ( ( ) ) can be controlled as follows: It follows from the product estimates (2.4) and the interpolation inequality (2.1) of Lemma 2.1 that and We can use the product estimates (2.4) to deduce and It follows through direct calculation that For ⩽ ⩽ l k 1 , we employ the integration by parts and the product estimates (2.4) to obtain which together with equation (4.18) yields easily that As a consequence of estimates (4.13)-(4.17)and estimate (4.19), it infers that 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.
Proof.Performing the operator ∇ l to the second equation of equation (4.1), the L 2 inner product of the resulting equation with ∇ ∇n For the first term ∫∇ ∂ ⋅∇ ∇ , it is derived directly by performing the integration by parts about the time variable t and the spatial variable We can take a similar approach to equation (3.40) to conclude By the product estimates (2.4), one can arrive at and similarly, Proof.Applying the operator ∇ l to the third equation of (4.1) and multiplying ∇ η l to the resulting identities, we obtain 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
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 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 We define k ℓ to be the expression under the time derivative in equation (4.30 For ⩽ ⩽ 0 1 ℓ , we have In the same manner, the remaining nonlinear terms can also be estimated and estimate (4.31) will be obtained.
For  The remaining terms can be obtained in a similar way and estimate (4.32) is proved.□

Proof of Theorem 1.3
In this subsection, we will prove Theorem 1.
It infers from the Hölder's inequality and the product estimates (2.4) that for = l 1,

45 )
Summing up equation (3.35) in Lemma 3.3 and equation (3.38) in Lemma 3.4 from 1 to − k 1, it infers 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

(4. 32 )
Proof.Under the conditions of Theorem 1.3, we can use the Hölder's inequality and the interpolation inequality (

ℓ
Adopting a similar approach, it can be obtained by direct calculation that .15)by the fact (2.6) of Lemma 2.3.We can perform the similar approach to estimates (4.13)-(4.15) to conclude Decay estimates of Navier-Stokes-Smoluchowski equations  17

25 )
Decay estimates of Navier-Stokes-Smoluchowski equations  19We can deal with the remaining nonlinear terms like equations (3.43) and (3.44), which combined with estimates (4.23)-(4.25)proveestimate(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.
From the decay property in Proposition 4.1, together with estimates (4.31) and (4.32) of the nonlinear terms, we can obtain