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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. doi:10.1098/rsta.2001.0852.

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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_0^1| 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 (v = 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. This is one article from the Discussion Meeting Issue ‘Topological methods in the physical sciences’ organized by V. I. Arnold, J. W. Bruce, H. K. Moffatt and R. B. Pelz.
Subject Keywords: Navier-Stokes; averaging; large-scale row; Euler equations; turbulence
Issue or Number:1784
Record Number:CaltechAUTHORS:20101005-112356720
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Official Citation:Marsden, J. E. and S. Shkoller (2001). "Global well–posedness for the Lagrangian averaged Navier–Stokes (LANS–α) equations on bounded domains." Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 359(1784): 1449-1468.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20305
Deposited By: Tony Diaz
Deposited On:16 Nov 2010 23:46
Last Modified:08 Nov 2021 23:58

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