Ill-posedness for the Incompressible Euler Equations in Critical Sobolev Spaces
By Tarek M. Elgindi, In-Jee Jeong. 2016
68 citations recorded by the source corpus.
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 $$L^\infty \cap H^1$$L∞∩H1 but escapes $$H^1$$H1 immediately for $$t>0$$t>0. Our main observation is that a localized chunk of vorticity bounded in $$L^\infty \cap H^1$$L∞∩H1 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.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5ff23f1cca89569f6faba356e9e0bde5a24b8396
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In Applied Mathematics
- Vlad Vicol
- T. Buckmaster
- Maria Colombo
- Martina Hofmanov'a
- Elia Brué
- Xiangchan Zhu
- Rongchan Zhu
- Dallas Albritton
In Mathematical Physics
- T. Buckmaster
- Vlad Vicol
- Maria Colombo
- Elia Brué
- Martina Hofmanov'a
- Nader Masmoudi
- Xiangchan Zhu
- Rongchan Zhu
Citations in the corpus
- Convex integration and phenomenologies in turbulence (2019, background): “In 2D, vorticity is transported, 15In some critical spaces, the Euler equations are known to be ill-posedness in the sense of Hadamard [10, 63, 61]. leading to global well-posedness of smooth solutions [96, 203] as well as weak solutions with L∞ bounded vorticity [205, 141, 198].”