Global Solutions to the Lagrangian Averaged Navier-Stokes equation in low regularity Besov spaces

TL;DR

This paper proves the existence of global solutions to the Lagrangian Averaged Navier-Stokes equations for initial data in a broad class of low regularity Besov spaces, extending previous results.

Contribution

It introduces an interpolation-based method to establish global solutions for initial data in Besov spaces with any p>3, broadening the known regularity conditions.

Findings

Global solutions exist for initial data in B^{3/p}_{p,q} with p>3.

Extends previous results from Sobolev and Besov spaces.

Uses interpolation techniques for proof.

Abstract

The Lagrangian Averaged Navier-Stokes (LANS) equations are a recently derived approximation to the Navier-Stokes equations. Existence of global solutions for the LANS equation has been proven for initial data in the Sobolev space $H^{3/4,2}(\mathbb{R}^3)$ and in the Besov space $B^{3/2}_{2,q}(\mathbb{R}^3)$. In this paper, we use an interpolation based method to prove the existence of global solutions to the LANS equation with initial data in $B^{3/p}_{p,q}(\mathbb{R}^3)$ for any $p>3$.