Local and global existence for the Lagrangian Averaged Navier-Stokes equations in Besov spaces

TL;DR

This paper establishes short-term existence of solutions for the Lagrangian Averaged Navier-Stokes equations in Besov spaces with low regularity initial data, and demonstrates global solutions under specific conditions in higher dimensions.

Contribution

It extends existence results for these equations to Besov spaces and improves upon Sobolev space results by allowing lower regularity initial data and higher dimensions.

Findings

Short-time solutions exist in Besov spaces for low regularity data.

Global solutions are obtained for certain parameters when p=2 and n≥3.

Advances previous Sobolev space results by relaxing regularity requirements.

Abstract

We prove the existence of short time solutions to the incompressible, isotropic Lagrangian Averaged Navier-Stokes equation with low regularity initial data in Besov spaces $B^{r}_{p,q}(\mathbb{R}^n)$, $r>n/2p$. When $p=2$ and $n\geq 3$, we obtain global solutions, provided the parameters $r,q$ and $n$ satisfy certain inequalities. This is an improvement over known analogous Sobolev space results, which required $n=3$.