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.

We prove higher-order and a Gevrey class (spatial analytic) regularity of
solutions to the Euler-Voigt inviscid $\alpha$-regularization of the
three-dimensional Euler equations of ideal incompressible fluids. Moreover, we
establish the convergence of strong solutions of the Euler-Voigt model to the
corresponding solution of the three-dimensional Euler equations for inviscid
flow on the interval of existence of the latter. Furthermore, we derive a
criterion for finite-time blow-up of the Euler equations based on this inviscid
regularization.

In this paper we develop and use successive averaging methods for explaining
the regularization mechanism in the the periodic Korteweg-de Vries (KdV
equation in the homogeneous Sobolev spaces $\dot{H}^s$, for $s \ge 0$.
Specifically, we prove the global existence, uniqueness, and Lipschitz
continuous dependence on the initial data of the solutions of the periodic KdV.
For the case where the initial data is in $L_2$ we also show the Lipschitz
continuous dependence of these solutions with respect to the initial data as
maps from $\dot{H}^s$ to $\dot{H}^s$, for $s\in(-1,0]$.

We present an alpha-regularization of the Birkhoff-Rott equation, induced by
the two-dimensional Euler-alpha equations, for the vortex sheet dynamics. We
show the convergence of the solutions of Euler-alpha equations to a weak
solution of the Euler equations for initial vorticity being a finite Radon
measure of fixed sign, which includes the vortex sheets case.