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

Abstract

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 $\alpha +1=\frac{n}{p}+\frac{2}{q}$ and $0<\alpha <2$. We prove the global in time existence of weak solution $u\in L^q(0,\infty ;{\dot{B}}_{pq}^{\alpha }({\mathbb{R}}_{+}^n))\cap L^{q_0}(0,\infty ;L^{p_0}({\mathbb{R}}_{+}^n))$ for some $1<p_0,q_0<\infty$ with $\frac{n}{p_0}+\frac{2}{q_0}=1$, when the given initial velocity has small norm in function space ${\dot{B}}_{p_0{q}_0,0}^{-\frac{2}{q_0}}({\mathbb{R}}_{+}^n)$. The solution is unique in the class $L^{q_0}(0,\infty ;L^{p_0}({\mathbb{R}}_{+}^n))$. Pressure estimates are also given.