Global well-posedness of the half space problem of the Navier-Stokes equations in critical function spaces of limiting case

Abstract

In this paper, we study the initial-boundary value problem of the Navier-Stokes equations in half-space. Let a solenoidal initial velocity be given in the function space $ \dot{B}_{p\infty,0}^{ -1 + n/p}({\mathbb R}^n_+)$ for $ \frac{n}3< p < n$. We prove the global in time existence of weak solution $u\in L^\infty(0,\infty; \dot B^{-1 +n/p}_{p\infty}({\mathbb R}^n_+))$, when the given initial velocity has small norm in function space $ \dot{B}_{p\infty,0}^{-1 + n/p} ({\mathbb R}^n_+)$, where $ \frac{n}3< p< n$.