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

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 $n=3$.