  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.
