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

Journal article

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 ∈ Hs ∩ H01 | Au = 0 on δΩ, 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 (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.

Authors: Marsden, J; Shkoller, S

• Journal: Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
• Volume: 359
• Issue: 1784
• Pages: 1449-1468
• Publication date: 2001-07-15
• DOI: 10.1098/rsta.2001.0852
• EISSN: 1471-2962
• ISSN: 1364-503X
• Language: English
• Keywords: Large-scale flow; Navier-Stokes; Averaging; Euler equations; Turbulence