Besov Regularity for the Stationary Navier-Stokes Equation on Bounded Lipschitz Domains

TL;DR

This paper investigates the regularity of solutions to the stationary Navier-Stokes and Stokes equations on Lipschitz domains using Besov spaces, which informs the convergence of approximation schemes.

Contribution

It introduces a Besov space framework for analyzing regularity of Navier-Stokes solutions on Lipschitz domains, extending previous Sobolev-based results.

Findings

Regularity results in Besov spaces for Stokes equations

Extension of analysis to Navier-Stokes equations using fixed point theorem

Implications for convergence rates of nonlinear approximation schemes

Abstract

We use the scale $B^s_{\tau}(L_\tau(\Omega))$, $1/\tau=s/d+1/2$, $s>0$, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains $\Omega\subset\mathbb{R}^d$, $d\geq 3$, with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. By using Banach's fixed point theorem, we extend this analysis to the stationary Navier-Stokes equation with suitable Reynolds number and data, respectively.