Local well-posedness for a free boundary problems for the compressible Navier--Stokes equations in critical spaces
Local well-posedness of the compressible Navier--Stokes equations with a free boundary condition is considered in the scaling critical spaces. We prove local well-posedness for the Lagrangetransformed compressible Navier--Stokes system in the homogeneous Besov space $(¥rho, u)¥in ¥dot B_{p,1}^{n/p}(R^n_+)¥times ¥dot B_{p,1}^{-1+n/p}(R^n_+)$ with the exponent $n-1< p< 2n-1$ along Solonnikov's formulation. To show local well-posedness, we use end-point maximal $L^1$-regularity for the corresponding linear initial-boundary value problem of the Lam¥'e equations corresponding to the velocity derived by the explicit Fourier symbols under the free boundary condition.
This is a joint work with Takayoshi Ogawa (Waseda University).