Note on the Euler equations in C^k spaces

2014/05/13 by Tarek M. Elgindi, Nader Masmoudi, Elgindi, Tarek M. +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions

paper · pdf · doi:10.48550/arxiv.1405.7891

openalex publication_date 2014/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, using the ideas from our recent article \citeEM, 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 \citeBLi2. 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: ∂_t D^k u=D^k+1 p + l.o.t. and then use the non-locality of the map u→ 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.

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