Strong solutions to the Cauchy problem of the two-dimensional compressible Navier-Stokes-Smoluchowski equations with vacuum

TL;DR

This paper establishes the local existence of strong solutions to the 2D compressible Navier-Stokes-Smoluchowski equations with vacuum, using weighted approximation techniques to overcome challenges specific to the two-dimensional case.

Contribution

It introduces a novel framework of weighted approximation estimates to prove local existence of strong solutions in 2D, extending previous results from 1D and 3D cases.

Findings

Local existence of strong solutions with vacuum in 2D.

Initial density can have compact support.

Extension of previous 1D and 3D results.

Abstract

This paper studies the local existence of strong solutions to the Cauchy problem of the 2D fluid-particle interaction model with vacuum as far field density. Notice that the technique used by Ding et al.\cite{SBH} for the corresponding 3D local well-posedness of strong solutions fails treating the 2D case, because the $L^p$-norm ($p>2$) of the velocity $u$ cannot be controlled in terms only of $\sqrt{\rho}u$ and $\nabla u$ here. In the present paper, we will use the framework of weighted approximation estimates introduced in [J. Li, Z. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl. (2014) 640--671] for Navier-Stokes equations to obtain the local existence of strong solutions provided the initial density and density of particles in the mixture do not decay very slowly at infinity. In particular, the initial density can have a compact support. This paper extends Fang et al.'s result \cite{DRZ} and Ding et al.'s result \cite{SBH}, in which, the existence is obtained when the space dimension $N=1$ and $N=3$ respectively.