Nowhere-differentiability of the solution map of 2D Euler equations on bounded spatial domain

Abstract

We consider the incompressible 2D Euler equations on bounded spatial domain $S$, and study the solution map on the Sobolev spaces $H^k(S)$ ($k>2$). Through an elaborate geometric construction, we show that for any $T>0$, the time $T$ solution map $u_0\mapsto u(T)$ is nowhere locally uniformly continuous and nowhere Fréchet differentiable.

14 pages