Global solutions to the generalized Leray-alpha equation with mixed dissipation terms. (May 2016) - Pennington, Nathan
Abstract: 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 − α 2 L 1 ) u for a Fourier Multiplier L ) and the generalized Navier–Stokes equation (which replaces the viscosity term ν △ with ν L 2 ). 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 L p ( R n ) based Sobolev space, the existence of a unique local solution with γ 1 + γ 2 > n / p + 1 . In the p = 2 case, the local solution is extended to a global solution, improving on previously known results.