Local and global low-regularity solutions to generalized Leray-alpha equations

Abstract

It has recently become common to study approximating equations for the Navier-Stokes equation. One of these is the Leray-alpha equation, which regularizes the Navier-Stokes equation by replacing (in most locations) the solution u with $(1-\alpha^2\Delta)u$. Another is the generalized Navier-Stokes equation, which replaces the Laplacian with a Fourier multiplier with symbol of the form $-|\xi|^\gamma$ ($\gamma=2$ is the standard Navier-Stokes equation), and recently in [16] Tao also considered multipliers of the form $-|\xi|^\gamma/g(|\xi|)$, where g is (essentially) a logarithm. The generalized Leray-alpha equation combines these two modifications by incorporating the regularizing term and replacing the Laplacians with more general Fourier multipliers, including allowing for g terms similar to those used in [16]. Our goal in this paper is to obtain existence and uniqueness results with low regularity and/or non-L^2 initial data. We will also use energy estimates to extend some of these local existence results to global existence results.

Keywords

• Leray-alpha model
• Besov space
• fractional Laplacian