Global Well-posedness for the Lagrangian Averaged Navier-Stokes (LANS-$\alpha$) Equations on Bounded Domains

Marsden, J. E. and S. Shkoller

Philos. Trans. Roy. Soc. London Ser. A, 359, 1449-1468

Abstract:

We prove the global well-posedness and regularity of the (isotropic) Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on a three-dimensional bounded domain with a smooth boundary with no-slip boundary conditions for initial data in the set $\{u \in H^s \cap H^1_0 \mid Au = 0 \text{ on }\partial\Omega, \operatorname{div}\, u = 0\}$, s $\in$ [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 $\alpha$-radius phase-space ball, and converge to the Navier-Stokes equations as $\alpha$$\to$ 0. We also show that classical solutions of the LANS-$\alpha$ equations converge almost all in H^s for s $\in$ (2.5, 3), to solutions of the inviscid equations ($\nu$ = 0), called the Lagrangian averaged Euler (LAE-$\alpha$) equations, even on domains with boundary, for time-intervals governed by the time of existence of solutions of the LAE-$\alpha$ equations.