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arXiv · 1603.07820

Ill-posedness for the incompressible Euler equations in critical Sobolev spaces

Tarek Mohamed Elgindi·In-Jee Jeong
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Abstract

For the $2D$ Euler equation in vorticity formulation, we construct localized smooth solutions whose critical Sobolev norms become large in a short period of time, and solutions which initially belong to $L^\infty \cap H^1$ but escapes $H^1$ immediately for $t>0$. Our main observation is that a localized chunk of vorticity bounded in $L^\infty \cap H^1$ with odd-odd symmetry is able to generate a hyperbolic flow with large velocity gradient at least for a short period of time, which stretches the vorticity gradient.

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Tarek Mohamed Elgindi, In-Jee Jeong. 2016-03-25. Ill-posedness for the incompressible Euler equations in critical Sobolev spaces. https://arxiv.org/abs/1603.07820

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