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

TL;DR

This paper analyzes how quickly solutions of two-dimensional $oldsymbol{ extalpha}$-models of turbulence converge to the true Navier-Stokes solutions as the regularization parameter approaches zero, providing explicit error estimates.

Contribution

It provides explicit convergence rates and error estimates for $oldsymbol{ extalpha}$-models and Galerkin approximations toward Navier-Stokes solutions in 2D with periodic boundary conditions.

Findings

Convergence rates are established in the $L^ ext{infty}$-$L^2$ norm as $oldsymbol{ extalpha} o 0$.

Error estimates for Galerkin approximations of the Leray-$oldsymbol{ extalpha}$ model are derived.

The results facilitate understanding of the accuracy of $oldsymbol{ extalpha}$-models in simulating 2D turbulence.

Abstract

Rates of convergence of solutions of various two-dimensional $\alpha-$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 $ \alpha$, when $\alpha$ approaches zero. Furthermore, as a paradigm, error estimates for the Galerkin approximation of the exact two-dimensional Leray-$\alpha$ 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 $\alpha$-regularization models toward the exact solutions of the Navier-Stokes equations in the two-dimensional periodic boundary conditions case.