Exponential growth of the vorticity gradient for the Euler equation on the torus

By Andrej Zlatoš. 2013

123 citations recorded by the source corpus.

We prove that there are solutions to the Euler equation on the torus with $C^{1,\alpha}$ vorticity and smooth except at one point such that the vorticity gradient grows in $L^\infty$ at least exponentially as $t\to\infty$. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Sverak.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a21ae827c243b20325e961c5c1ea4a7a8408ece2

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In Applied Mathematics

1. Tarek M. Elgindi
2. Andrej Zlatoš
3. In-Jee Jeong
4. Yao Yao
5. Lenya Ryzhik
6. Alexander V. Kiselev
7. Vlad Vicol
8. Peter Constantin

In Geometry and Topology

1. Tarek M. Elgindi
2. In-Jee Jeong

Citations in the corpus

• Finite time singularity for the modified SQG patch equation (2016, background): “Exponential growth on a domain without a boundary (the torus T2) was recently shown to be possible by Zlatoš [56].”
• Ill-posedness for the Incompressible Euler Equations in Critical Sobolev Spaces (2016, methodology): “Certain perturbations of this stationary solution were utilized in the works of Denissov [5], Kiselev-Šverák [8], and Zlatoš [10] (in chronological order) to obtain growth of vorticity gradient in the maximum norm L∞.”