Local and Global low-regularity solutions to the Generalized Leray-alpha equations

Abstract

It has recently become common to study many different approximating equations of 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 $u$ in the equation with $(1-\alpha^2\triangle)u$ the operator $(1-\alpha^2\triangle)$. 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 [14] 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 [14]. 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.