Local well-posedness for the initial boundary value problem of the scaling critical compressible Navier–Stokes equations in nearly half space

We consider the initial boundary value problem for the compressible Navier-Stokes equations of barotropic type. It is well-known that the equations maintain its structure nearly invariant under the scaling invariance and by the Fujita–Kato principle, the problem can be solved in the scaling invariant homogeneous Besov space locally in time for the Cauchy problem. We show the initial boundary value problem of the system with $ 0 $-Dirichlet boundary condition has a unique local solution in the nearly half Euclidean space. We employ the Lagrangian transform to transfer the equation in the Lagrangian coordinate, and flattening the boundary function to the half space and show the local well-posedness of the system. The key for the proof is employing end-point $ L^1 $ maximal regularity for the parabolic equations in half space shown in [1].