$L_1$ approach to the compressible viscous fluid flows in the half-space

TL;DR

This paper establishes local well-posedness for compressible viscous Navier-Stokes equations in a half-space, using an $L_1$ framework and semigroup methods, with solutions in Besov spaces.

Contribution

It introduces an $L_1$ in time approach combined with Besov space regularity for analyzing compressible Navier-Stokes in a half-space, utilizing Lagrange transformation and analytic semigroup techniques.

Findings

Proved local well-posedness in the half-space setting.

Developed an $L_1$ maximal regularity framework for the Stokes semigroup.

Established regularity of density and velocity solutions in Besov spaces.

Abstract

In this paper, we proved the local well-posedness for the Navier-Stokes equtions describing the motion of isotropic barotoropic compressible viscous fluid flow with non-slip boundary conditions, wehre the fluid domain is the half-space in the $N$-dimensional Euclidean space. The density part of solutions and their time derivative belong to $L_1$ in time with some Besov spaces in space and also the velosity parts and their time derivative belong to $L_1$ in time with some Besov spaces in space. We use Lagrange transformation to eliminate the covection term and we use an analytic semgroup approach. Our Stokes semigroup is not only a continuous analytic semigroup but also has an $L_1$ in times maximal regularity with some Besov spaces in space.