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-$\alpha$) 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.<br> 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.