Anomalous dissipation for 1/5-Hölder Euler flows

Tristan Buckmaster, Camillo De Lellis, Philip Isett, László Székelyhidi

Research output: Contribution to journalArticlepeer-review

86 Scopus citations

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 ε, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the Hölder class C1/5-ε.

Original languageEnglish (US)
Pages (from-to)127-172
Number of pages46
JournalAnnals of Mathematics
Volume182
Issue number1
DOIs
StatePublished - 2015

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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