On the nonlinear stability of shear flows and vortices

Abstract

In this article we present some of the main ideas in our recent work on the asymptotic stability of shear flows and vortices among solutions of the Euler equations in two dimensions. More precisely, we discuss the following results:

1. a theorem on the nonlinear asymptotic stability of a large class of shear flows $(b(y),0)$ in the finite channel $\mathbb{T}\times [0,1]$, defined by strictly increasing Gevrey smooth functions $b$, which are linear outside a compact subset of the interval $(0,1)$ and satisfy suitable spectral conditions;
2. a theorem on the nonlinear asymptotic stability of point vortex solutions of the Euler equation in ${\mathbb{R}}^2$;
3. heuristic analysis showing that the mechanism of inviscid damping is unlikely to work to produce global solutions of the $\alpha$-generalized SQG equation in two dimensions, for any parameter $\alpha >0$.