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

TL;DR

This paper demonstrates that the solution map for 2D Euler equations on bounded domains is nowhere locally uniformly continuous and nowhere Fréchet differentiable, highlighting complex geometric properties of fluid flow solutions.

Contribution

It provides a novel geometric construction proving the solution map's lack of differentiability and uniform continuity for all positive times.

Findings

Solution map is nowhere locally uniformly continuous.

Solution map is nowhere Fréchet differentiable.

Results hold for any positive time T.

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 Fr\'echet differentiable.