an:06568218
Zbl 1334.76035
Pennington, Nathan
Global solutions to the generalized Leray-alpha equation with mixed dissipation terms
EN
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 136, 102-116 (2016).
00354539
2016
j
76D05 35A01
Leray-alpha model; fractional Laplacian; global existence
Summary: 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-\(\alpha\) equation (which replaces the solution \(u\) with \((1 - \alpha^2 \mathcal{L}_1) u\) for a Fourier Multiplier \(\mathcal{L}\)) and the generalized Navier-Stokes equation (which replaces the viscosity term \(\nu \triangle\) with \(\nu \mathcal{L}_2\)). In this paper we consider the combination of these two equations, called the generalized Leray-\(\alpha\) equation. We provide a brief outline of the typical strategies used to solve such equations, and prove, with initial data in a low-regularity \(L^p(\mathbb{R}^n)\) based Sobolev space, the existence of a unique local solution with \(\gamma_1 + \gamma_2 > n / p + 1\). In the \(p = 2\) case, the local solution is extended to a global solution, improving on previously known results.