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The Lagrangian averaged Navier-Stokes equation with rough data in Sobolev spaces. (English) Zbl 1284.35348
Summary: The Lagrangian Averaged Navier-Stokes equation is a recently derived approximation to the Navier-Stokes equation. In this article we prove the existence of short time solutions to the incompressible, isotropic Lagrangian Averaged Navier-Stokes equation with low regularity initial data in Sobolev spaces \(W^{s,p}(\mathbb R^n)\) for \(1<p<\infty\). For \(L^2\)-based Sobolev spaces, we obtain global existence results. More specifically, we achieve local existence with initial data in the Sobolev space \(H^{n/2p,p}(\mathbb R^n)\). For initial data in \(H^{3/4,2}(\mathbb R^3)\), we obtain global existence, improving on previous global existence results, which required data in \(H^{3,2}(\mathbb R^3)\).

MSC:
35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35A01 Existence problems for PDEs: global existence, local existence, non-existence
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