Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations

Modelling the mean characteristics of turbulent channel flow has been one of the longstanding problems in fluid dynamics. While a great number of mathematical models have been proposed for isotropic turbulence, there are relatively few, if any, turbulence models in the anisotropic wall-bounded regime which hold throughout the entire channel. Recently, the anisotropic Lagrangian averaged Navier-Stokes equations (LANS-α) have been derived in [7] and [5]. This paper is devoted to the analysis of this coupled system of nonlinear PDE for the mean velocity and covariance tensor in the channel geometry. The vanishing of the covariance along the walls induces certain degenerate elliptic operators into the model, which require weighted Sobolev spaces to study. We prove that when the no-slip boundary conditions are prescribed for the mean velocity, the LANS-α equations possess unique global weak solutions which converge as time tends to infinity towards the unique stationary solutions. Qualitative properties of the stationary solutions are also established