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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.

##### MSC:
 76D05 Navier-Stokes equations for incompressible viscous fluids 35A01 Existence problems for PDEs: global existence, local existence, non-existence
##### Keywords:
Leray-alpha model; fractional Laplacian; global existence
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##### References:
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