Anomalous dissipation and Euler flows

Jan Burczak; László Székelyhidi; Bian Wu

Anomalous dissipation and Euler flows

Analysis of PDEs 2024-09-19 v2

Authors: Jan Burczak , László Székelyhidi , Bian Wu

Abstract

We show anomalous dissipation of scalars advected by weak solutions to the incompressible Euler equations with $C^{(\sfrac{1}{3})^-}$ regularity, for an arbitrary initial datum in $\dot H^1 (\T^3)$ . This is the first rigorous derivation of zeroth law of scalar turbulence, where the scalar is advected by solution to an equation of hydrodynamics (unforced and deterministic). As a byproduct of our method, we provide a typicality statement for the drift, and recover certain desired properties of turbulence, including a lower bound on scalar variance commensurate with the Richardson pair dispersion hypothesis.

Keywords

euler equations euler equations for incompressible fluids fluid dynamics

Cite

@article{arxiv.2310.02934,
  title  = {Anomalous dissipation and Euler flows},
  author = {Jan Burczak and László Székelyhidi and Bian Wu},
  journal= {arXiv preprint arXiv:2310.02934},
  year   = {2024}
}

Comments

Updated version 66 pages. The new version includes an extended introduction discussing implications for and the relationship to hydrodynamic turbulence, and a more general statement of the main result. Some minor typos were also corrected

Anomalous dissipation and Euler flows

Abstract

Keywords

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