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For any $A > 2$, we construct solutions to the two-dimensional incompressible Euler equations on the torus $\mathbb{T}^2$ whose vorticity gradient $\nabla\omega$ grows exponentially in time: $$\|\nabla\omega(t, \cdot)\|_{L^\infty} \gtrsim e^{At},\quad \forall\ t \geq 0.$$
E. Holder, U ¨ {\rm\ddot{U}} ber die unbeschr a ¨ {\rm\ddot{a}} nkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkompressiblen Fl u ¨ {\rm\ddot{u}} ssigkeit
1933
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W. Wolibner, Un theor e ` {\rm\grave{e}} me sur l’existence du mouvement plan d’un uide parfait, homog e ` {\rm\grave{e}} ne, incompressible, pendant un temps infiniment long (French)
1933
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V. I. Yudovich, The flow of a perfect, incompressible liquid through a given region
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J.-Y. Chemin, Perfect Incompressible Fluids
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A. Majda and A. Bertozzi, Vorticity and Incompressible Flow
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T. Tao, http://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard
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S. Denisov, Infinite superlinear growth of the gradient for the two-dimensional Euler equation
2009
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J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schr o ¨ {\rm\ddot{o}} dinger equation
2010
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J. Bourgain, Personal communications
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S. Denisov, Double-exponential growth of the vorticity gradient for the two-dimensional Euler equation
Cited in the paper.
Cited in the paper.
A. Kiselev and V. Šverák, Small scale creation for solutions of the incompressible two dimensional Euler equation
2013
Later among the works it cites.
J. Bourgain and D. Li, Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces. Invent. Math. 201 (2015), no. 1, 97–157
2015
Later among the works it cites.
A. Zlatoš, Exponential growth of the vorticity gradient for the Euler equation on the torus
2015
Later among the works it cites.
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