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 ∞-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 presented in the L ∞-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.