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We are interested in the stability analysis of two-dimensional incompressibleinviscid fluids. Specifically, we revisit a recent result on the stability ofYudovich's solutions to the incompressible Euler equations in$L^\infty([0,T];H^1)$ by providing a new approach to its proof based on theidea of compactness extrapolation and by extending it to the whole plane.  This new method of proof is robust and, when applied to viscous models, leadsto a remarkable logarithmic improvement on the rate of convergence in thevanishing viscosity limit of two-dimensional fluids. Loosely speaking, thislogarithmic gain is the result of the fact that, in appropriate high-regularitysettings, 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 ``memoryeffect'' seems to be a general principle which is not exclusive to fluidmechanics. It is therefore likely to be observed in other setting and deservesfurther investigation.  Finally, we also apply the stability results on Euler systems to the study oftwo-dimensional ideal plasmas and establish their convergence, in strongtopologies, to solutions of magnetohydrodynamic systems, when the speed oflight tends to infinity. The crux of this asymptotic analysis relies on a fineunderstanding of Maxwell's system.

Stability analysis of two-dimensional ideal flows with applications to viscous fluids and plasmas