An Onsager singularity theorem for Leray solutions of incompressible Navier-Stokes

Theodore D. Drivas, Gregory L. Eyink

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus Td, assuming that the solutions have norms for Besov space Bσ,&inf; 3 (Td), σ &insin; (0, 1], that are bounded in the L3-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form O(v(3σ-1)/(σ+1)), vanishing as v ← 0 if σ > 1/3. A consequence is that Onsager-type 'quasi-singularities' are required in the Leray solutions, even if the total energy dissipation vanishes in the limit v ← 0, as long as it does so sufficiently slowly. We also give two sufficient conditions which guarantee the existence of limiting weak Euler solutions u which satisfy a local energy balance with possible anomalous dissipation due to inertial-range energy cascade in the Leray solutions. For σ &insin; (1/3, 1) the anomalous dissipation vanishes and the weak Euler solutions may be spatially 'rough' but conserve energy.

Original languageEnglish (US)
Pages (from-to)4465-4482
Number of pages18
JournalNonlinearity
Volume32
Issue number11
DOIs
StatePublished - Oct 10 2019

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Keywords

  • anomalous dissipation
  • fluid turbulence
  • Onsager's conjecture

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