  We are interested in the stability analysis of two-dimensional incompressible
inviscid fluids. Specifically, we revisit a recent result on the stability of
Yudovich's solutions to the incompressible Euler equations in
$L^\infty([0,T];H^1)$ by providing a new approach to its proof based on the
idea of compactness extrapolation and by extending it to the whole plane.
  This new method of proof is robust and, when applied to viscous models, leads
to a remarkable logarithmic improvement on the rate of convergence in the
vanishing viscosity limit of two-dimensional fluids. Loosely speaking, this
logarithmic gain is the result of the fact that, in appropriate high-regularity
settings, the smoothness of solutions to the Euler equations at times $t\in
[0,T)$ is strictly higher than their regularity at time $t=T$. This ``memory
effect'' seems to be a general principle which is not exclusive to fluid
mechanics. It is therefore likely to be observed in other setting and deserves
further investigation.
  Finally, we also apply the stability results on Euler systems to the study of
two-dimensional ideal plasmas and establish their convergence, in strong
topologies, to solutions of magnetohydrodynamic systems, when the speed of
light tends to infinity. The crux of this asymptotic analysis relies on a fine
understanding of Maxwell's system.
