# American Institute of Mathematical Sciences

March  2002, 1(1): 35-50. doi: 10.3934/cpaa.2002.1.35

## Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains

 1 Department of Mathematics, University of California, Davis, CA 95616, United States

Received  April 2001 Published  December 2001

In this paper, we study the Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations on bounded domains. The LANS-$\alpha$ equations are able to accurately reproduce the large-scale motion (at scales larger than $\alpha >0$) of the Navier-Stokes equations while filtering or averaging over the motion of the fluid at scales smaller than α, an a priori fixed spatial scale.
We prove the global well-posedness of weak $H^1$ solutions for the case of no-slip boundary conditions in three dimensions, generalizing the periodic-box results of [8]. We make use of the new formulation of the LANS-$\alpha$ equations on bounded domains given in [20] and [14], which reveals the additional boundary conditions necessary to obtain well-posedness. The uniform estimates yield global attractors; the bound for the dimension of the global attractor in 3D exactly follows the periodic box case of [8]. In 2D, our bound is $\alpha$-independent and is similar to the bound for the global attractor for the 2D Navier-Stokes equations.
Citation: Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
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