Global in time solvability of the Navier-Stokes equations in the half-space

In this paper, we study the initial value problem of the Navier-Stokes equations in the half-space. Let a solenoidal initial velocity be given in the function space $ \dot{B}_{pq,0}^{\alpha-\frac{2}{2}}({\mathbb R}^n_+)$ for $\alpha +1 = \frac{n}p + \frac2q$ and $0<\alpha<2$. We prove the global in time existence of weak solution $u\in L^q(0,\infty; \dot B^\alpha_{pq}({\mathbb R}^n_+))\cap L^{q_0}(0, \infty; L^{p_0}({\mathbb R}^n_+)) $ for some $ 1<p_0, q_0<\infty$ with $\frac{n}{p_0} +\frac2{q_0} =1$, when the given initial velocity has small norm in function space $ \dot{B}_{p_0q_0,0}^{-\frac{2}{q_0}}({\mathbb R}^n_+)$. The solution is unique in the class $L^{q_0}(0, \infty; L^{p_0}({\mathbb R}^n_+))$. Pressure estimates are also given.