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

TL;DR

This paper investigates how solutions of the inviscid Euler-$\alpha$ model for complex fluids converge to classical Euler solutions as the regularization parameter diminishes, providing convergence rates and conditions for existence.

Contribution

It establishes convergence in Sobolev spaces and estimates the convergence rate for 2D vortex patches with smooth boundaries.

Findings

Convergence in $H^{s}$ for $s>n/2+1$ in the whole space.

Existence of smooth Euler-$\alpha$ solutions at least as long as Euler solutions.

Quantitative convergence rate for 2D vortex patches.

Abstract

We study the convergence rate of the solutions of the incompressible Euler-$\alpha$, an inviscid second-grade complex fluid, equations to the corresponding solutions of the Euler equations, as the regularization parameter $\alpha$ approaches zero. First we show the convergence in $H^{s}$, $s>n/2+1$, in the whole space, and that the smooth Euler-$\alpha$ 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.