Solvability of the Initial-Boundary value problem of the Navier-Stokes equations with rough data

In this paper, we study the initial and boundary value problem of the Navier-Stokes equations in the half space. We prove the unique existence of weak solution $u\in L^q({\mathbb{R}}_{+}\times (0,T))$ with $\nabla u\in L^{\frac{q}{2}}{loc}(\R+\times (0,T))$ for a short time interval when the initial data $h\in B_q^{-\frac{2}{q}}({\mathbb{R}}_{+})$ and the boundary data $ g\in L^q(0,T;B^{-\frac{1}{q}}_q(\Rn))+L^q(\Rn;B^{-\frac{1}{2q}}_q(0,T)) $ with normal component $g_n\in L^q(0,T;{\dot{B}}_q^{-\frac{1}{q}}(\text{Rn}))$, $n+2<q<\mathrm{\infty }$ are given.