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

TL;DR

This paper proves the global existence and uniqueness of weak solutions to the Navier-Stokes equations in a half-space for small initial data in specific Besov spaces, extending understanding of fluid flow in unbounded domains.

Contribution

It establishes global in time solvability of Navier-Stokes in the half-space with initial data in Besov spaces, including pressure estimates, for the first time under these conditions.

Findings

Global weak solutions exist for small initial data.

Solutions are unique in certain Lebesgue space classes.

Pressure estimates are provided for the solutions.

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 $\alpha +1 = \frac{n}p + \frac2q$ 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.