On the Extension of Onsager’s Conjecture for General Conservation Laws

Journal of Nonlinear Science - Tập 29 - Trang 501-510 - 2018
Claude Bardos1, Piotr Gwiazda2, Agnieszka Świerczewska-Gwiazda3, Edriss S. Titi4,5, Emil Wiedemann6
1Laboratoire J.-L. Lions, Paris Cedex 05, France
2Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland
3Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland
4Department of Mathematics, Texas A&M University, College Station, USA
5Department of Computer Science and Applied Mathematics, The Weizmann Institute of Science, Rehovot, Israel
6Institute of Applied Analysis, Universität Ulm, Ulm, Germany

Tóm tắt

The aim of this work is to extend and prove the Onsager conjecture for a class of conservation laws that possess generalized entropy. One of the main findings of this work is the “universality” of the Onsager exponent, $$\alpha > 1/3$$ , concerning the regularity of the solutions, say in $$C^{0,\alpha }$$ , that guarantees the conservation of the generalized entropy, regardless of the structure of the genuine nonlinearity in the underlying system.

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