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

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 $C^{\alpha, \frac12 \alpha} ({\mathbb R}^n_+ \times (0, T)), 0 < \alpha < 1$, when the given initial data is in $C^\alpha ({\mathbb R}^n_+)$ and the given boundary data is in $C^{\alpha, \frac12 \alpha} ({\mathbb R}^{n-1} \times (0, T))$. Our result generalizes the result in [30] considering nonhomogeneous Dirichlet boundary data.