Solvability of the Initial-Boundary value problem of the Navier-Stokes equations with rough data

TL;DR

This paper establishes the existence and uniqueness of weak solutions to the Navier-Stokes initial-boundary value problem in a half space with rough initial and boundary data, extending the solvability theory to less regular data.

Contribution

It proves the solvability of the Navier-Stokes equations with rough data in the half space, including new conditions on initial and boundary data for weak solutions.

Findings

Unique weak solutions exist for short time with rough data

Solutions belong to specific Lebesgue and Besov space classes

The results extend previous solvability conditions to less regular data

Abstract

In this paper, we study the initial and boundary value problem of the Navier-Stokes equations in the half space. We prove the unique existence of weak solution $u\in L^q(\R_+\times (0,T))$ with $\nabla u\in L^{\frac{q}{2}}_{loc}(\R_+\times (0,T))$ for a short time interval when the initial data $h\in {B}_q^{-\frac{2}{q}}(\R_+)$ and the boundary data $ g\in L^q(0,T;B^{-\frac{1}{q}}_q(\Rn))+L^q(\Rn;B^{-\frac{1}{2q}}_q(0,T)) $ with normal component $g_n\in L^q(0,T;\dot{B}^{-\frac{1}{q}}_q(\Rn))$, $n+2<q<\infty$ are given.