Initial-Boundary value problem of the Navier-Stokes equations in the half space with nonhomogeneous data

This paper discusses the solvability (global in time) of the initial-boundary value problem of the Navier-stokes equations in the half space when the initial data $ h\in \dot{ B}_{q \sigma}^{\alpha-\frac{2}{q}}(\R_+)$ and the boundary data $ g\in \dot{ B}_q^{\alpha-\frac{1}{q},\frac{\al}{2}-\frac{1}{2q}}({\mathbb R}^{n-1}\times {\mathbb R}_+) $ with $g_n\in \dot B^{\frac12 \alpha}_q ({\mathbb R}_+; \dot B^{-\frac1q}_q ({\mathbb R}^{n-1}))\cap L^q({\mathbb R}_+;\dot{B}^{\alpha-\frac{1}{q}}(\Rn))$, for any $0<\alpha<2$ and $q =\frac{n+2}{\alpha+1}$. Compatibility condition is required for $h$ and $g$.