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-α equation (which replaces the solution u with (1-^α2 _L1)u for a Fourier Multiplier L) and the generalized Navier-Stokes equation (which replaces the viscosity term νδ with ν_L2). In this paper we consider the combination of these two equations, called the generalized Leray-α 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) based Sobolev space, the existence of a unique local solution with _γ1+_γ2>n/p+1. In the p=2 case, the local solution is extended to a global solution, improving on previously known results.