The Lagrangian Averaged Navier–Stokes equation with rough data in Sobolev spaces

Abstract

The Lagrangian Averaged Navier–Stokes equation is a recently derived approximation to the Navier–Stokes equation. In this article we prove the existence of short time solutions to the incompressible, isotropic Lagrangian Averaged Navier–Stokes equation with low regularity initial data in Sobolev spaces Ws,p(Rn) for 1<p<∞. For L2-based Sobolev spaces, we obtain global existence results. More specifically, we achieve local existence with initial data in the Sobolev space Hn/2p,p(Rn). For initial data in H3/4,2(R3), we obtain global existence, improving on previous global existence results, which required data in H3,2(R3).