This lecture will discuss the relevance of the Euler fluid equations to the description of high Reynolds-number turbulent flows, emphasizing recent progress and important open issues. We first succinctly review empirical observations on "anomalous energy dissipation" coming from laboratory experiments and numerical simulations and then discuss Onsager's conjecture that connects the phenomenon to weak/singular/distributional solutions of the Euler equations. Recent work of De Lellis & Székelyhidi Jr. and collaborators has yielded existence of weak Euler solutions with spatial Hölder regularity and with anomalous dissipation of kinetic energy pointwise in space-time. This work also shows, however, that such Euler solutions are non-unique for a dense set of finite-energy initial conditions. This non-uniqueness raises the question whether additional "admissibility conditions'' exist which can select unique physically-relevant Euler solutions. The obvious condition is that such Euler solutions should arise from the zero-viscosity limit of Navier-Stokes solutions, but so far only much weaker solutions (measure-valued solutions of DiPerna-Majda, "dissipative solutions" of Lions) have been so obtained. Anomalous dissipation in this inviscid limit requires finite-time singularities
for smooth Euler solutions. The widely accepted physical ideas of G. I. Taylor on stretching of vortex-lines make a doubtful assumption of conservation of circulations in conventional form for turbulent flow, but a recent stochastic martingale formulation of the Kelvin Theorem by Constantin-Iyer resolves this problem. Further insight has been obtained from exactly soluble model problems, particularly Burgers equation and the Kraichnan passive-scalar model. An extremely significant discovery in the latter by Bernard-Gaw?dzki-Kupiainen is the "spontaneous stochasticity" of Lagrangian particle trajectories due to Richardson super-diffusive dispersion and its relation to anomalous dissipation. In the opinion of the speaker, one of the most important outstanding problems of theoretical turbulence is to understand the connection between spontaneous stochasticity and anomalous dissipation for Navier-Stokes turbulence
in both two and three (and higher) space dimensions.
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