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

It has recently become common to study approximating equations for the Navier-Stokes equation. One of these is the Leray-α equation, which regularizes the Navier-Stokes equation by replacing (in most locations) the solution u with (1 -α²Δ)u. Another is the generalized Navier-Stokes equation, which replaces the Laplacian with a Fourier multiplier with symbol of the form -|ξ|^γ (γ = 2 is the standard Navier-Stokes equation), and recently in [16] Tao also considered multipliers of the form -|ξ|^γ /g(|ξ|), where g is (essentially) alogarithm. The generalized Leray-a 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-L² initial data. We will also use energy estimates to extend some of these local existence results to global existence results.