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

Marsden, Jerrold E. and Shkoller, Steve (2001) Global well–posedness for the Lagrangian averaged Navier–Stokes (LANS–α) equations on bounded domains. Philosophical Transactions A: Mathematical, Physical and Engineering Sciences, 359 (1784). pp. 1449-1468. ISSN 1364-503X.

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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 {u ∈ H^s ∩ H^1_0| 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 H^s 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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Additional Information:© 2001 The Royal Society. We thank The Royal Society for the opportunity to present this work at their interesting Discussion Meeting. We thank Darryl Holm for his kind suggestions and remarks on earlier drafts of this paper. We also thank Daniel Coutand and James Peirce for carefully reading the manuscript and making many valuable suggestions for its improvement. J.E.M. and S.S. were partly supported by the NSF-KDI grant ATM-98-73133. J.E.M. also acknowledges the support of the California Institute of Technology and S.S. was partly supported by the Alfred P. Sloan Foundation Research Fellowship.
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Alfred P. Sloan FoundationUNSPECIFIED
Subject Keywords:Navier-Stokes; averaging; large-scale flow; Euler equations; turbulence
Issue or Number:1784
Record Number:CaltechAUTHORS:20100909-085510720
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19840
Deposited By: Ruth Sustaita
Deposited On:09 Sep 2010 17:14
Last Modified:10 Sep 2020 23:06

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