Note on the Euler equations in C^k spaces

TL;DR

This paper proves strong ill-posedness of the 2D Euler equations in $C^k$ spaces, showing that derivatives can develop singularities immediately, using a shorter proof and analyzing the pressure term's non-local effects.

Contribution

Provides a shorter proof of ill-posedness for 2D Euler equations in $C^k$ spaces and demonstrates immediate singularity formation in derivatives.

Findings

Existence of initial data with immediate derivative singularities.

Ill-posedness extends to $C^{k-1,1}$ spaces.

Ill-posedness driven by pressure term's non-locality.

Abstract

In this note, using the ideas from our recent article \cite{EM}, we prove strong ill-posedness for the 2D Euler equations in $C^k$ spaces. This note provides a significantly shorter proof of many of the main results in \cite{BLi2}. In the case $k>1$ we show the existence of initial data for which the $kth$ derivative of the velocity field develops a logarithmic singularity immediately. The strong ill-posedness covers $C^{k-1,1}$ spaces as well. The ill-posedness comes from the pressure term in the Euler equation. We formulate the equation for $D^k u$ as: $$\partial_t D^k u=D^{k+1} p + l.o.t.$$ and then use the non-locality of the map $u\rightarrow p$ to get the ill-posedness. The real difficulty comes in how to deal with the "l.o.t." terms which can be handled by special commutator estimates.