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.

Uniform time of existence for the alpha Euler equations

“…Because of the identities ∇ = − n n 2 × (n × ∇) + n n 2 (∂ n ) and u = − n n 2 × (n × u) + n n 2 (n · u) we observe that it suffices to bound ∂ n (n · u) W k,∞ co and ∂ n (n × u) W k,∞ co . Thanks to (10) we have that…”

Global generalized solutions for Maxwell-alpha and Euler-alpha equations

“…solutions to the Euler problem as α → 0. The convergence is known for the strong solutions on the time interval where they exist [24].…”

Local well‐posedness and zero‐αlimit for the Euler‐αequations

“…In [12], the global existence of weak solutions is shown for the 3D axisymmetric Euler-˛equations without swirl, for initial vorticity being a finite Radon measure with compact support; moreover, the global existence and uniqueness is established for the compactly supported vorticity in L p , with p > 3 2 . The convergence rate of the solutions of the incompressible Euler-˛equations to the corresponding solutions of the Euler equations results for the initial value problem associated with (1.1) was first proved by Linshiz and Titi [13] for initial data v0 2 H s .R d / with s > d 2 C 1. It has been recently shown that the solutions of the Euler-˛system converge to the corresponding solution of the Euler equations as the presence of boundary layer in the norm of L 2 .…”