On the rate of convergence of the two-dimensional $α$-models of turbulence to the Navier-Stokes equations

Abstract

Rates of convergence of solutions of various two-dimensional $α-$regularization models, subject to periodic boundary conditions, toward solutions of the exact Navier-Stokes equations are given in the $L^\infty$-$L^2$ time-space norm, in terms of the regularization parameter $ α$, when $α$ approaches zero. Furthermore, as a paradigm, error estimates for the Galerkin approximation of the exact two-dimensional Leray-$α$ model are also presented in the $L^\infty$-$L^2$ time-space norm. Simply by the triangle inequality, one can reach the error estimates of the solutions of Galerkin approximation of the $α$-regularization models toward the exact solutions of the Navier-Stokes equations in the two-dimensional periodic boundary conditions case.

29 pages