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Global well–posedness for the Lagrangian averaged Navier–Stokes (LANS–α) equations on bounded domains

J. E. Marsden, S. Shkoller

Abstract

We prove the global well–posedness and regularity of the (isotropic) Lagrangian averaged Navier–Stokes (LANS–α) equations on a three–dimensional bounded domain with a smooth boundary with no–slip boundary conditions for initial data in the set {uHs ∩ H10| Au = 0 on ∂Ω, div u = 0}, s ∈ [3, 5), where A is the Stokes operator. As with the Navier–Stokes equations, one has parabolic–type regularity; that is, the solutions instantaneously become space–time smooth when the forcing is smooth (or zero). The equations are an ensemble average of the Navier–Stokes equations over initial data in an α–radius phase–space ball, and converge to the Navier–Stokes equations as α → 0. We also show that classical solutions of the LANS–α equations converge almost all in Hs for s ∈ 2.5, 3), to solutions of the inviscid equations (ν = 0), called the Lagrangian averaged Euler (LAE–α) equations, even on domains with boundary, for time–intervals governed by the time of existence of solutions of the LAE–α equations.

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