$L_1$ approach to the compressible viscous fluid flows in general domains

TL;DR

This paper establishes $L_1$ maximal regularity for the Stokes equations in general domains with $C^3$ boundaries, enabling well-posedness results for compressible Navier-Stokes equations without domain compactness assumptions.

Contribution

It introduces a new method based on real interpolation to achieve $L_1$ maximal regularity, removing the need for domain compactness in prior studies.

Findings

Proves $L_1$ in time maximal regularity for Stokes equations in general domains.

Establishes local well-posedness for compressible Navier-Stokes equations with Dirichlet boundary conditions.

Develops a versatile method applicable to parabolic and hyperbolic-parabolic systems with non-homogeneous boundary conditions.

Abstract

We prove the $L_1$ in time and $B^{s+1}_{q,1}\times B^s_{q,1}$ in space maximal regularity for the Stokes equations in the viscous compressible fluid flows in domains in the $N$ dimensional Euclidean space $R^N$ whose boundary is $C^3$ compact hypersurface. As an application, the local well-posedness is proved for the compressible Navier-Stokes equations with Dirichlet condition. Danchin and Tolksdorf have studied the same problem in a bounded domains. Since they used Da-Prato and Grisvard theory directly, they need the assumption that the domain is compact. We imvestigate a new method to obtain $L_1$ maximal regularity which is based on real interpolation theories and thanks to this method, we can remove the compactness assumptions in the study due to Danchin and Torksdorf. Our method can be applied to obtain $L_1$ in time maximal regularity theorem for the initial boundary value problem of the system of parabolic or hyperbolic-parabolic equations with non-homogeneous boundary conditions.