Calculus of Variations and Geometric Measure Theory
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T. Buckmaster - C. De Lellis - L. J. Sz\'ekelyhidi

Dissipative Euler flows with Onsager-critical spatial regularity

created by delellis on 05 May 2014
modified on 07 Mar 2015

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Inserted: 5 may 2014
Last Updated: 7 mar 2015

Year: 2014

Abstract:

For any $\varepsilon >0$ we show the existence of continuous periodic weak solutions $v$ of the Euler equations which do not conserve the kinetic energy and belong to the space $L^1_t (C_x^{1/3-\varepsilon})$, namely $x\mapsto v (x,t)$ is $(1/3-\varepsilon)$-H\"older continuous in space at a.e. time $t$ and the integral $\int [v(\cdot, t)]_{1/3-\varepsilon}\, dt$ is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class $L^\infty_t (C_x^{1/3-\varepsilon})$


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