On the convergence rate of the Euler-$α$, an inviscid second-grade complex fluid, model to the Euler equations

Abstract

We study the convergence rate of the solutions of the incompressible Euler-$α$, an inviscid second-grade complex fluid, equations to the corresponding solutions of the Euler equations, as the regularization parameter $α$ approaches zero. First we show the convergence in $H^{s}$, $s>n/2+1$, in the whole space, and that the smooth Euler-$α$ solutions exist at least as long as the corresponding solution of the Euler equations. Next we estimate the convergence rate for two-dimensional vortex patch with smooth boundaries.