Anomalous dissipation for $1/5$-Hölder Euler flows

Abstract

Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces. The motivation comes from Onsager’s conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field.
In a recent paper the third author has improved upon the methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent — albeit weaker than the one conjectured by Onsager. In this paper we give a shorter proof of this final result, adhering more to the original scheme of the second and fourth authors and introducing some new devices. More precisely we show that for any positive $\varepsilon$, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the Hölder class $C^{1/5-\varepsilon}$.

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Authors

Tristan Buckmaster

Institut für Mathematik, Universität Leipzig, Leipzig, Germany

Current address:

Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 Camillo De Lellis

Institut für Mathematik, Universität Zürich, Zürich, Winterthurerstrasse 190, CH-190 ü, Switzerland

Philip Isett

Department of Mathematics, Princeton University, Princeton, NJ

Current address:

Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139 László Székelyhidi, Jr.

Institut für Mathematik, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany