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

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