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

Abstract

In this paper, we study the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains. The LANS-α equations are able to accurately reproduce the large-scale motion (at scales larger than α> 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-α 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 α-independent and is similar to the bound for the global attractor for the 2D Navier-Stokes equations.