Initial and boundary value problem of the unsteady Navier–Stokes system in the half-space with Hölder continuous boundary data

Abstract

We study the initial–boundary value problem of the Navier–Stokes system in the half-space. We prove the unique solvability of the weak solution for some short time interval (0,T) with the velocity in Cα,α2(R+n×(0,T)), 0<α<1, when the given initial data in Cα(R+n) and the given boundary data in Cα,α2(Rn−1×(0,T)) satisfy the compatibility conditions. Our result generalizes the result in [29] considering nonhomogeneous Dirichlet boundary data.