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Global solutions to the generalized Leray-alpha equation with mixed dissipation terms. (English) Zbl 1334.76035
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.

76D05 Navier-Stokes equations for incompressible viscous fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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