On the regularity of the solution map of the incompressible Euler equation

TL;DR

This paper proves that the solution map of the incompressible Euler equations in Sobolev spaces is nowhere locally uniformly continuous and nowhere differentiable, highlighting irregularity in the solution dependence on initial data.

Contribution

It establishes the irregularity of the solution map for the Euler equations in Sobolev spaces, showing it is nowhere locally uniformly continuous and nowhere differentiable.

Findings

Solution map is nowhere locally uniformly continuous.

Solution map is nowhere differentiable.

Irregular dependence on initial data in Sobolev spaces.

Abstract

In this paper we consider the incompressible Euler equation on the Sobolev space $H^s(\R^n)$, $s > n/2+1$, and show that for any $T > 0$ its solution map $u_0 \mapsto u(T)$, mapping the initial value to the value at time $T$, is nowhere locally uniformly continuous and nowhere differentiable.