GLOBAL EXISTENCE AND TIME-DECAY ESTIMATES OF SOLUTIONS TO THE COMPRESSIBLE NAVIER-STOKES-SMOLUCHOWSKI EQUATIONS

. This paper is concerned with the Cauchy problem of the compressible Navier-Stokes-Smoluchowski equations in R 3 . Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces, the existence theory of global solutions in H 3 to the stationary proﬁle is established. Moreover, when the initial perturbation is bounded in L p -norm with 1 ≤ p < 65 , we obtain the optimal convergence rates of the solution in L q -norm with 2 ≤ q ≤ 6 and its ﬁrst order derivative in L 2 -norm.

The fluid-particle interaction model plays an important role in sedimentation analysis of disperse suspensions of particles in fluids. Its applications in biotechnology, medicine, chemical engineering, mineral processes, et. al can be referred for instance to [3,5,19,21,22]. The system (1.1) consists in a Vlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled to the Navier-Stokes equations for a compressible fluid. Without the dynamic viscosity terms in (1.1) 2 , this system was derived formally by Carrillo and Goudon [6]. There are two different scaling limits for the coupling system between the kinetic and the fluid equations: the so-called bubbling and flowing regimes. They correspond to the diffusive approximation of the kinetic equation, the bubbling regime, written in (1.1), and the strong drag force and strong Brownian motion for the flowing regime (Refer to [6,7] for more details). In [6], they considered the flowing regime and the bubbling regime under the two different scaling assumptions and investigated the stability and asymptotic limits finally. There have been some known results on the local and global well-posedness of the solutions of (1.1) in one dimension ( [11,18]). 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 [4], Carrillo et. al's work [7] and Ballew's work [2], respectively. In particular, Carrillo et. al in their work [7] prove that the weak solutions exist globally in time and that the weak solutions converge to a stationary solution as time goes to ∞. More precisely, Carrillo et. al derived the following two theorems: Theorem A (Carrillo-Karper-Trivisa: Global Existence). Assume that (Ω, Φ) satisfy the "confinement hypotheses". Then, the problem (1.1) supplemented with boundary conditions u| ∂Ω = (∇ x η · ν + η∇ x Φ · ν)| ∂Ω = 0 and initial data ρ(x, 0) = ρ 0 ∈ L γ (Ω)∩L 1 + (Ω), (ρu)(x, 0) = m 0 ∈ L 6 5 (Ω)∩L 1 (Ω), η(x, 0) = η 0 ∈ L 2 (Ω)∩L 1 + (Ω) admits a weak solution (ρ, u, η) on Ω × (0, ∞).
Theorem B (Carrillo-Karper-Trivisa: Large-Time Asymptotics). Assume that (Ω, Φ) satisfy the "confinement hypotheses". Then, for any free energy solution (ρ, u, η) of the problem (1.1), in the sense of "Definition 2.2", there exist universal stationary states ρ s (x), η s (x), such that as t → ∞. However, how fast does the solution converge to the stationary solutions as t → ∞? It is still open. This motivates us to study the time-decay rates of the solutions.
Similar to the statement about the stationary solution of Navier-Stokes equations in [17], for system (1.1), there exists a stationary solution (ρ * , u * , η * )(x) in a small neighborhood of (ρ ∞ , 0, 0) such that and that for some positive constant C. Here Φ H 3 ≤ for some positive constant as in (2.6).
2. Preliminary and main results. Before stating the main results, we would like to give a reformulation of (1.1). More precisely, denote Then the initial value problem (1.1) is reformulated as as |x| → ∞, To simplify the computations in the proof, we denote Then by (1.2), (2.1) can be rewritten as where 3) and , The initial data consequently becomes Our main purpose in the paper is to study the global existence and time-decay rates of the solution (ρ, u, η) in a small perturbation of the stationary solution (ρ * , 0, 0), i.e., the existence and decay rates of the perturbed solution ( , u, z). In what follows, we state our main results. The first one is concerned with the global existence of the solution.
, then there exists a constant C 0 independent of t, such that Remark 2.3. Compared with the stationary state in [7], we consider a special case of the stationary state of η for simplicity, i.e., η * (x) = 0. However, here we obtain the time-decay rate of the solution near the stationary state (ρ * , 0, 0).
Remark 2.4. In Theorem 2.2, we prove that the solution (ρ, u, η) of (1.1) converges to the stationary solution (ρ * , 0, 0) at some time speed, and that the decay speeds in time for the zero-order spatially derivative and the first-order spatially derivative are different. For the study of the stability in time of more general stationary solutions, we leave it in the near future.
3. Global existence. Define the solution space of the initial value problem (2.1) with a norm as follows: for any 0 ≤ T ≤ +∞. The following is the local existence of the solution to the problem (2.2).
Remark 3.1. Proposition 3.1 is a special case of Theorem 2.1 in [9]. Thus, the proof is omitted here for simplicity. Refer also to [14,17] for the ideas of the proof.
In what follows, we will establish some a priori estimates of the solutions globally in time. With the help of the local existence theory and those estimates, the global existence of solutions will be obtained by employing the standard continuity argument. To begin with, we make a priori assumption for some T ∈ (0, T * ) where T * represents the maximal time of existence of the solutions, and the constant δ sufficiently small is chosen in (3.48). Using the Sobolev imbedding inequality, we are able to obtain that and 3) Here "· ·" represents that "· ≤ C · " for some known constant C > 0.
With the a priori assumption (3.1), we obtain the following estimates which can ensure the global existence of the solution. The first one is the L 2 estimate of ( , u).
Proof. Multiplying (2.2) 1 , (2.2) 2 by and u, respectively, and then integrating by parts over R 3 , we have from the sum of the resulting equalities that (3.5) Using integration by parts and the Young inequality, we have To estimate the last two terms on the right-hand side of (3.5), we notice that the source terms S 1 and S 2 have the following equivalent properties under the conditions of (2.6) and (3.1): With the help of (3.7), we use the Hölder inequality, Lemma 4.3, (1.2), (2.6) and the Young inequality, and then obtain where we also have used the following Hardy inequality With (3.8), similar to the proof of , S 1 , we get Thus, we complete the proof of Lemma 3.1.
Proof. Multiplying ∇ k (2.2) 1 , ∇ k (2.2) 2 by ∇ k and ∇ k u, respectively, and then integrating by parts over R 3 , we have from the sum of the resultant equalities that (3.12) We are going to estimate the terms on the right hand side of the above equality. More precisely, for the first term on the right hand side of (3.12), we get For the second term on the right hand side of (3.12), we obtain (3.14) For I 1 , we have where we have used the Hölder inequality, (3.1) and (4.40).
For I 2 , we have where we have used the Hölder inequality, (3.1) and (4.41). Similar to the evaluations of I 1 and I 2 , for I 3 and I 4 , we get  For the third term on the right hand side of (3.12), we obtain For J 1 , we have and and With those estimates above, we complete the proof of the lemma.
Proof. Applying ∇ k to (2.2) 2 and then taking the L 2 inner product with ∇∇ k , we have (3.26) With (2.2) 1 , the first term on the right hand side of (3.26) is estimated as follows: (3.28)

YINGSHAN CHEN, SHIJIN DING AND WENJUN WANG
The second term and the third term on the right hand side of (3.26) can be estimated as follows: where the Cauchy inequality has been used twice. With (3.8), ∇ k S 2 L 2 on the fourth term of the right hand side of (3.26) can be estimated as follows. and and and for k = 0, 1, 2, 3.
Proof. Taking ∇ k over (2.2) 3 , Multiplying the resulting equations by 2∇ k z, and integrating by parts over R 3 , we have We are going to evaluate the terms on the right hand side of (3.37) term by term. More precisely, we have holds for any 1 ≤ p < 6 5 . Remark 3.2. The time-decay estimate of z(t) L 2 in (3.40) will be used to get a time-independent bound of t 0 z(s) 2 L 2 ds. It plays a very important role to close the a priori assumption (3.1). For more details, please refer to (3.48).
Proof. It follows from (2.2) 4 and the classical theory of linear parabolic equations (refer for instance to [15,23]) that z is given by where L(t) : φ → v(·, t) is the solution semigroup defined by L(t) = e −t∆ , i.e., v(x, t) = L(t)φ = K(·, t) * φ(·), Notice that K(t) L 1 = 1 and and that This combined with the smallness of δ and implies that The proof of Lemma 3.5 is complete. Now we are in a position to close the a priori assumption (3.1). From (3.4), (3.11), and (3.25), for a fixed small constant 1 > 0, we have d dt (3.44) With (3.44) and the smallness of and δ, we have d dt where Then, from (3.35), we have From (3.45) and (3.46), we get d dt Using (3.40) and (3.47), we have (3.48) Here we choose δ > max{ 3 which is the desired estimate for proving that the maximal time for existence T * = ∞. The proof of Theorem 2.1 is completed.
4. Decay rates. The proof of Theorem 2.2 is divided into two subsections.

4.1.
Decay rates of linearized system. We consider the Cauchy problem of the corresponding linearized system of (2.2), i.e., The solution ( , u, z) of the linearized system (4.1) can be expressed as Here G(t) := G(x, t) is the Green matrix of the system. Similar to the proof in [12], we have the following lemma.
We recall the following estimates, cf. [13,20]. Here, we give the proof of Lemma 5.4 for the convenience of readers. Then we have and [∇ m , f ]g L p ∇f L p 1 ∇ m−1 g L p 2 + ∇ m f L p 3 g L p 4 , (4.41) where p, p 2 , p 3 ∈ (1, +∞) and Proof.