Initial-boundary value problem of the Navier-Stokes system in the half space

Abstract

In this paper, 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 on some short time interval (0, T) with the velocity in $C^{\alpha ,\frac{1}{2}\alpha }({\mathbb{R}}_{+}^n\times (0,T)),0<\alpha <1$, when the given initial data is in $C^{\alpha }({\mathbb{R}}_{+}^n)$ and the given boundary data is in $C^{\alpha ,\frac{1}{2}\alpha }({\mathbb{R}}^{n-1}\times (0,T))$. Our result generalizes the result in [30] considering nonhomogeneous Dirichlet boundary data.