Double exponential growth of the vorticity gradient for the two-dimensional Euler equation

Denisov, Sergey

doi:10.1090/S0002-9939-2014-12286-6

Double exponential growth of the vorticity gradient for the two-dimensional Euler equation
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by Sergey A. Denisov PDF

Proc. Amer. Math. Soc. 143 (2015), 1199-1210 Request permission

Abstract:

For the two-dimensional Euler equation on the torus, we prove that the $L^\infty$–norm of the vorticity gradient can grow as double exponential over arbitrary long but finite time provided that at time zero it is already sufficiently large. The method is based on the perturbative analysis around the singular stationary solution studied by Bahouri and Chemin in 1994. Our result on the growth of the vorticity gradient is equivalent to the statement that the operator of Euler evolution is not bounded in the linear sense in Lipschitz norm for any time $t>0$.

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