Global solutions to the generalized Leray-alpha equation with mixed dissipation terms

Due to the intractability of the Navier–Stokes equation, it is common to study approximating equations. Two of the most common of these are the Leray-αα equation (which replaces the solution uu with (1−α2L1)u(1−α2L1)u for a Fourier Multiplier LL) and the generalized Navier–Stokes equation (which replaces the viscosity term ν△ν△ with νL2νL2). In this paper we consider the combination of these two equations, called the generalized Leray-αα equation. We provide a brief outline of the typical strategies used to solve such equations, and prove, with initial data in a low-regularity Lp(Rn)Lp(Rn) based Sobolev space, the existence of a unique local solution with γ1+γ2>n/p+1γ1+γ2>n/p+1. In the p=2p=2 case, the local solution is extended to a global solution, improving on previously known results.