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

Theodore D. Drivas, Gregory L. Eyink

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


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
Issue number11
StatePublished - Oct 10 2019

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics


  • Onsager's conjecture
  • anomalous dissipation
  • fluid turbulence


Dive into the research topics of 'An Onsager singularity theorem for Leray solutions of incompressible Navier-Stokes'. Together they form a unique fingerprint.

Cite this