Initial-Boundary value problem of the Navier-Stokes equations in the half space with nonhomogeneous data

TL;DR

This paper establishes the global solvability of the Navier-Stokes initial-boundary value problem in a half space with nonhomogeneous data within specific Besov spaces, under certain compatibility conditions.

Contribution

It provides new results on the existence of solutions for Navier-Stokes equations with nonhomogeneous boundary and initial data in Besov spaces, extending previous work to broader data classes.

Findings

Global in time solvability established

Solutions exist under specific Besov space conditions

Compatibility conditions are necessary for data

Abstract

This paper discusses the solvability (global in time) of the initial-boundary value problem of the Navier-stokes equations in the half space when the initial data $ h\in \dot{ B}_{q \sigma}^{\alpha-\frac{2}{q}}(\R_+)$ and the boundary data $ g\in \dot{ B}_q^{\alpha-\frac{1}{q},\frac{\al}{2}-\frac{1}{2q}}({\mathbb R}^{n-1}\times {\mathbb R}_+) $ with $g_n\in \dot B^{\frac12 \alpha}_q ({\mathbb R}_+; \dot B^{-\frac1q}_q ({\mathbb R}^{n-1}))\cap L^q({\mathbb R}_+;\dot{B}^{\alpha-\frac{1}{q}}(\Rn))$, for any $0<\alpha<2$ and $q =\frac{n+2}{\alpha+1}$. Compatibility condition is required for $h$ and $g$.