On the $L_{1}$ maximal regularity in the study of free boundary problem for the compressible fluid flows

Abstract

In this paper, we consider the Stokes equations with non-homogeneous free boundary conditions, which are obtained by the linearization procedure of the free boundary problem of the Navier–Stokes equations describing the viscous compressible fluid flows. We prove the $L_{1}$ maximal regularity of solutions to these Stokes equations. This is an extension result of $L_{p}$-$L_{q}$ maximal regularity result obtained by D. Götz and Y. Shibata to the $L_{1}$ in time maximal regularity case.