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It has recently become common to study many different approximating equationsof 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), andrecently in [14] Tao also considered multipliers of the form$|\xi|^\gamma/g(|\xi|)$, where $g$ is (essentially) a logarithm. Thegeneralized Leray-$\alpha$ equation combines these two modifications byincorporating the regularizing term and replacing the Laplacians with moregeneral Fourier multipliers, including allowing for $g$ terms similar to thoseused in [14]. Our goal in this paper is to obtain existence and uniquenessresults with low regularity and/or non-$L^2$ initial data. We will also useenergy estimates to extend some of these local existence results to globalexistence results.

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