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

Abstract

In this paper, we prove the local well-posedness for the Navier-Stokes equations describing the motion of isotropic barotoropic compressible viscous fluid flow with non-slip boundary conditions, where the fluid domain is the $N$ dimensional half-sapce. We solve the equations in the $L_1$ in time and Besov spaces $B_{q,1}^s$ in space maximal regularity framework. Here, we assume that $-1+N/q\le s<1/q$ and $N-1<q<2N$. We use Lagrange transformation to eliminate the convection term and we use an analytic semigroup approach. We only assume the strictly positiveness of initial mass density.

This paper treats the $L_1$ maximal regularity for the compressible Stokes equations in the half space with Dirichlet zero condition and the local well-posedness of the compressible Navier-Stokes equations in the half space in the $L_1$ in time framework