The Lagrangian Averaged Navier Stokes (LANS) equations have
been introduced as surrogates for the Navier Stokes equations in the
simulation of turbulence in a periodic domain in R^3. In this talk I will provide some
of the motivation leading to these system of equations as well as results
obtained in collaboration with L Chen, R Guenther, S-C Kim and E Waymire on the rate
of convergence of solutions of the regularized equations to solutions of the
Navier-Stokes equations.