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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 $H^{1//2}$

Joel Avrin 1,

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.
Keywords: extensions of operators., integral equations, Lagrangian averaged Navier-Stokes, concentration mapping.
Mathematics Subject Classification: 35Q3.
Citation: Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure & Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353
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