R solver approach to the maximal regularity and free bounary problem for the Navier Stokes equations

Abstract

First, I will talk about the R solver method created by myself to prove the maximal regularity for the initial-boundary problem with non-homogeneous boundary date. By using this method, I proved the Lp in time and Lq in space maximal regularity theorem for the Stokes equations with free boundary conditions. The idea of R solver is based on Weis operator valued Fourier multiplier theorem. As an application, I will talk about the local and global wellposedness of free boundary problem of the Navier-Stokes equations.