Local well-posedness to the Cauchy problem of the 2D compressible Navier-Stokes-Smoluchowski equations with vacuum

Abstract

This paper concerns the local strong solution to the compressible Navier-Stokes-Smoluchowski equations with vacuum as far field density on the whole space ${\mathbb{R}}^2$ without additional Cho-Choe-Kim type compatibility conditions, which provided the initial density and the initial particles density decay not too slow at infinity. In particular, we extend the results of Ding et al. (2016) [13] and Yang (2020) [24] to the 2D case.

Introduction

In this paper, we consider a fluid-particle interaction model called as Navier-Stokes-Smoluchowski equations in [2], [7], [8], which in the whole spatial domain ${\mathbb{R}}^2$ as follows

$$\{\begin{array}{c}{\rho }_t+\mathrm{div}(\rho u)=0,\hfill \\ {(\rho u)}_t+\mathrm{div}(\rho u\otimes u)+\mathrm{\nabla }(p_F+\eta )=\mu \mathrm{\Delta }u+(\lambda +\mu )\mathrm{\nabla }\mathrm{div}u-(\eta +\beta \rho )\mathrm{\nabla }\mathrm{\Phi },\hfill \\ {\eta }_t+\mathrm{\nabla }\cdot (\eta (u-\mathrm{\nabla }\mathrm{\Phi }))=\mathrm{\Delta }\eta ,\hfill \end{array}$$

in ${\mathbb{R}}^2\times {\mathbb{R}}^{+}$, with the far-field behavior

$$(\rho ,u,\eta )(x,t)\to (0,0,0)\mathrm{as}|x|\to \infty ,t>0,$$

and initial data

$$\rho (x,0)={\rho }_0(x),\rho u(x,0)=m_0,\eta (x,0)={\eta }_0,x\in {\mathbb{R}}^2.$$

Here $\rho :{\mathbb{R}}^2\times [0,\infty )\to {\mathbb{R}}^{+}$ is the density of the fluid, $u:{\mathbb{R}}^2\times [0,\infty )\to {\mathbb{R}}^2$ the velocity field, and the density of the particles in the mixture $\eta :(0,\infty )\times {\mathbb{R}}^2\to {\mathbb{R}}_{+}$ is related to the probability distribution function $f(t,x,\xi )$ in the macroscopic description through the relation

$$\eta (t,x)=\underset{{\mathbb{R}}^2}{\int }f(t,x,\xi )d\xi .$$

We also denote by $p_F$ the pressure of the fluid, given by

$$p_F=p_F(\rho )=a{\rho }^{\gamma },a>0,\gamma >1,$$

and β is a constant reflecting the differences in how the external force affects the fluid and the particles, λ and μ are constant viscosity coefficients satisfying the physical condition:

$$\mu >0,\lambda +\mu \ge 0.$$

The time independent external potential $\mathrm{\Phi }=\mathrm{\Phi }(x):{\mathbb{R}}^2\to {\mathbb{R}}_{+}$ is the effects of gravity and buoyancy.

The fluid-particle interaction model arises in a lot of industrial procedures such as the analysis of sedimentation phenomenon which finds its applications in biotechnology, medicine, chemical engineering, and mineral processes. Such interaction systems are also used in combustion theory, when modeling diesel engines or rocket propulsors, see [5], [6], [22], [23]. The system consists in a Vlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled to the equations for the fluid. Generally speaking, at the microscopic scale, the cloud of particles is described by its distribution function $f(t,x,\xi )$, solution to a Vlasov-Fokker-Planck equation. The fluid, on the other hand, is modeled by macroscopic quantities, namely its density $\rho (x,t)\ge 0$ and its velocity field $u(x,t)$ (see [7]). If the fluid is compressible and isentropic, then $(\rho ,u)$ solves the compressible Euler (inviscid case) or Navier-Stokes system (viscous case) of equations. With the dynamic viscosity terms taken into consideration, system (1.1) was derived formally by Carrillo-Goudon [8]. They obtained the global existence and asymptotic behavior of the weak solutions to (1.1) following the framework of Lions [21] and Feireisl et al. [16], [17]. Without the dynamic viscosity terms in ${\text{(1.1)}}_2$, Carrillo-Goudon [7] gave the flowing regime and the bubbling regime under the two different scaling assumptions and investigated the stability and asymptotic limits finally. In dimension one, Fang et al. [15] proved the global existence and uniqueness of the classical large solution with vacuum. In dimension three, Ballew obtained the local in time existence of strong solutions in a bounded domain with the no-flux condition for the particle density in [2], [4] and studied Low Mach Number Limits under the confinement hypotheses for the spatial domain and external potential Φ in [3]. Recently, motivated by Kim et al. [9], [10], [11] on the Navier-Stokes equations, Ding et al. [13] obtained the local classical solutions of system (1.1) with vacuum in ${\mathbb{R}}^3$. However, even the local existence of strong solution to the two-dimensional problem (1.1)–(1.3) with vacuum is still unknown.

When the density of the fluid $\eta =0$, the system (1.1) becomes Navier-Stokes equations for the isentropic compressible fluids. Kim et al. proved some local existence results on strong solutions in a domain of ${\mathbb{R}}^3$ in [9], [11] and the radially symmetric solutions in an annular domain in [12]. Ding et al. [14] obtained global classical solutions with large initial data with vacuum in a bounded domain or exterior domain Ω of ${\mathbb{R}}^n(n\ge 2)$. In a bounded or unbounded domain of ${\mathbb{R}}^3$, Cho-Kim also got the local classical solutions [10], in which the initial density needs not be bounded below away from zero. For the case that the initial density is allowed to vanish, Huang et al. [18] obtained the global existence of classical solutions to the Cauchy problem for the isentropic compressible Navier-Stokes equations in three spatial dimensions with smooth initial data provided that the initial energy is suitably small. Recently, assumed that the initial density do not decay very slowly at infinity, Li-Liang [19] have obtained the local existence of the classical solutions to the two-dimensional Cauchy problem. After that, Li-Xin [20] extended the result of Li-Liang [19] to the global ones, and also get some decay estimates of solutions.

Throughout this paper, always denote

$$\overline{x}\triangleq {(\mathrm{e}+|x{|}^2)}^{1/2}{\log }^{1+{\sigma }_0}(\mathrm{e}+|x{|}^2),$$

with ${\sigma }_0>0$ fixed. The main result of this paper is stated as the following theorem:

Theorem 1.1

Suppose that the initial data $({\rho }_0,m_0,{\eta }_0)$ satisfy

$$\{\begin{array}{c}{\rho }_0\ge 0,{\overline{x}}^a{\rho }_0\in L^1({\mathbb{R}}^2)\cap H^1({\mathbb{R}}^2)\cap W^{1,q}({\mathbb{R}}^2),\hfill \\ \mathrm{\nabla }{\eta }_0\in L^2({\mathbb{R}}^2),\mathrm{\nabla }{u}_0\in L^2({\mathbb{R}}^2),\sqrt{{\rho }_0}{u}_0\in L^2({\mathbb{R}}^2),\hfill \\ {\overline{x}}^{\frac{a}{2}}{\eta }_0\in L^2({\mathbb{R}}^2),\mathit{\Phi }\in H^5({\mathbb{R}}^2),m_0={\rho }_0{u}_0,\hfill \end{array}$$

with $q>2$ and $a>1$. Then there exist $T_0,N>0$ such that the problem (1.1)–(1.3) has a unique strong solution $(\rho ,u,\eta )$ on ${\mathbb{R}}^2\times (0,T_0]$ satisfying

$$\{\begin{array}{c}\rho \in C([0,T_0];L^1({\mathbb{R}}^2)\cap H^1({\mathbb{R}}^2)\cap W^{1,q}({\mathbb{R}}^2)),\hfill \\ {\overline{x}}^a\rho \in L^{\infty }(0,T_0;L^1({\mathbb{R}}^2)\cap H^1({\mathbb{R}}^2)\cap W^{1,q}({\mathbb{R}}^2)),\hfill \\ \sqrt{\rho }u,\mathrm{\nabla }u,{\overline{x}}^{-1}u,\sqrt{t}\sqrt{\rho }{u}_t\in L^{\infty }(0,T_0;L^2({\mathbb{R}}^2)),\hfill \\ \mathrm{\nabla }u\in L^2(0,T_0;H^1({\mathbb{R}}^2))\cap L^{\frac{q+1}{q}}(0,T_0;W^{1,q}({\mathbb{R}}^2)),\hfill \\ \sqrt{t}\mathrm{\nabla }u\in L^2(0,T_0;W^{1,q}({\mathbb{R}}^2)),\hfill \\ \eta ,\mathrm{\nabla }\eta ,{\overline{x}}^{\frac{a}{2}}\eta ,\sqrt{t}{\eta }_t\in L^{\infty }(0,T_0;L^2({\mathbb{R}}^2)),\hfill \\ \mathrm{\nabla }\eta \in L^2(0,T_0;H^1({\mathbb{R}}^2)),\sqrt{t}\mathrm{\nabla }u\in L^2(0,T_0,W^{1,q}({\mathbb{R}}^2)),\hfill \\ \sqrt{\rho }{u}_t,{\overline{x}}^{\frac{a}{2}}\mathrm{\nabla }\eta ,\sqrt{t}\mathrm{\nabla }{u}_t,\sqrt{t}\mathrm{\nabla }{\eta }_t,\sqrt{t}{\overline{x}}^{-1}{u}_t\in L^2({\mathbb{R}}^2\times (0,T_0)),\hfill \end{array}$$

and

$$\underset{0\le t\le T_0}{\inf }\underset{B_N}{\int }\rho (x,t)dx\ge \frac{1}{4}\underset{{\mathbb{R}}^2}{\int }{\rho }_0(x,t)dx.$$

Remark 1.1

It is worth noting that no Cho-Choe-Kim type compatibility condition (see [9], [10], [11]) on the initial data is required in Theorem 1.1 for the local existence and uniqueness of strong solutions.

Remark 1.2

Theorem 1.1 extends the results of Ding et al. [13] (also see Yang [24]) to the Cauchy problem in two dimensional (2D) space.

We now make some comments on the key ingredients of the analysis in this paper. Notice that the local well-posedness of strong solutions for dimension three case established by Ding et al. [13] is not admitted for the case of dimension two. This is mainly due to that in dimension two we fail to control the $L^p$-norm ($p>2$) of the velocity u in terms only of $\sqrt{\rho }u$ and ∇u. Moreover, the coupling of $u,\eta$ and Φ, and the presence of $\mathrm{\nabla }\cdot (\eta u-\eta \mathrm{\nabla }\mathrm{\Phi })$ bring additional difficulties.

In order to overcome these difficulties stated above, some new ideas and observations are needed. To deal with the difficulty caused by the unbounded domain, following the idea of [19] and in light of ρ decaying at infinity of x, we estimate the $L^p({\mathbb{R}}^2)$-norm for momentum ρu instead of the velocity u with ${\Vert {\rho }^{\zeta }u\Vert }_{L^{\sigma }}$ ($\sigma >\max \{2,2/\zeta \}$) controlled in terms of ${\Vert \sqrt{\rho }u\Vert }_{L^2}$ and ${\Vert \mathrm{\nabla }u\Vert }_{L^2}$, where the Hardy-type inequalities (cf. Lemma 2.2) are substantial for building these estimates. In contrast to [19], to overcome the difficulties caused by the coupling terms of $u,\eta$ and Φ, such as $\Vert |u||\eta |\Vert$ and $\Vert |u||\mathrm{\nabla }\eta |\Vert$, we deduced some spatial weighted estimate on η (i.e., ${\overline{x}}^a\eta$, see (3.13)). Next, we construct approximate solutions to (1.1), that is, for density strictly away from vacuum initially, we consider a initial boundary value problem of (1.1) in any bounded ball $B_R$ with radius $R>0$. Finally, combining all these ideas stated above, we derive some desired bounds on the gradients of the velocity and the spatial weighted ones on both the density and its gradients, which are independent of both the radius of the balls $B_R$ and the lower bound of the initial density.

The rest of the paper is organized as follows: In Section 2, we collect some elementary facts and inequalities as preliminaries. Section 3 is devoted to the a priori estimates which required for obtaining the local existence and uniqueness of strong solutions. Finally, the proof of Theorem 1.1 is given in Section 4.