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 $u\in L^{\infty }(0,\infty ;{\dot{B}}_{p\infty }^{-1+n/p}({\mathbb{R}}_{+}^n))$, when the given initial velocity has small norm in function space ${\dot{B}}_{p\infty ,0}^{-1+n/p}({\mathbb{R}}_{+}^n)$, where $\frac{n}{3}<p<n$.

arXiv admin note: text overlap with arXiv:1809.07025