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

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 $-1+N\mathrm{/}q\le s<1\mathrm{/}q$ 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.