Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^(1/2)$

Abstract

We consider the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on a bounded domain in $R^{3}$ with zero (no-slip) boundary conditions. With periodic boundary conditions on a box, these equations are also known as the Camassa-Holm equations. The (LANS-$\alpha$) model averages or coarse-grains the small, computationally unreasonable, scales of the Navier-Stokes equations; spatial scales smaller than $\alpha>0$ are averaged out. We establish the existence and uniqueness of local strong (i.e., regular) solutions with initial data in $H^{1//2}$, and then use the a priori estimate developed in [1] to conclude that these are global regular solutions. Our results extend those in [2] and [1], which show the global well-posedness of $H^{1}$ weak solutions in a periodic box and on a bounded domain with no-slip boundary conditions, respectively.