Compressible Navier-Stokes system on a moving domain in the $L_p-L_q$ framework

TL;DR

This paper establishes local and global well-posedness results for the compressible Navier-Stokes equations on a moving domain within an $L_p-L_q$ framework, allowing for broader conditions on the velocity field and its derivatives.

Contribution

It extends the analysis of compressible Navier-Stokes equations to moving domains with minimal restrictions on the velocity field, using maximal regularity techniques.

Findings

Proves local well-posedness in the $L_p-L_q$ framework.

Shows global existence under smallness and decay conditions.

Provides decay estimates for solutions.

Abstract

We prove the local well-posedness for the barotropic compressible Navier-Stokes system on a moving domain, a motion of which is determined by a given vector field ${\bf V}$, in a maximal $L_p-L_q$ regularity framework. Under additional smallness assumptions on the data we show that our solution exists globally in time and satisfies a decay estimate. In particular, for the global well-posedness we don't require exponential decay or smallness of ${\bf V}$ in $L_p(L_q)$. However, we require exponential decay and smallness of its derivatives.