# American Institute of Mathematical Sciences

September  2004, 3(3): 353-366. doi: 10.3934/cpaa.2004.3.353

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

 1 Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, United States

Received  January 2004 Revised  March 2004 Published  June 2004

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.
Citation: Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure and Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353
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