Anomalous Dissipation, Spontaneous Stochasticity and Onsager’s Conjecture — Two Tales

Theodore Drivas
Johns Hopkins University
Applied Mathematics and Statistics

This talk is divided into two parts. The first part discusses Lagrangian spontaneous stochasticity, or the non-uniqueness of Lagrangian particle trajectories at infinite Reynolds number, which was discovered in work on the Kraichnan model of turbulent scalar advection to be intimately related to dissipative anomalies for advected scalars. There has been some debate whether this mechanism is realistic and applies, in particular, when the advecting velocity field is a solution of incompressible Navier-Stokes equation. We settle the issue rigorously for domains without boundaries (e.g. tori, spheres), and for wall-bounded flows with no-flux Neumann conditions for the scalar, where we prove that spontaneous stochasticity is necessary and sufficient for anomalous scalar dissipation with any advecting velocity field whatsoever. The proof exploits a “Lagrangian fluctuation-dissipation relation” (LFDR) for scalars, both passive and active. The LFDR has other applications, e.g. to Nusselt-number scaling in turbulent Rayleigh-Bénard convection. The derivation of the LFDR shows that the scalar field evaluated along stochastic Lagrangian trajectories is a martingale backward in time. I discuss a conjecture of Eyink that such a martingale property is an admissibility condition for the limiting weak solutions conjectured by Onsager to exist at infinite Reynolds and Péclet numbers. The second part of the talk discusses a development of Onsager’s theory of turbulence for compressible fluids. We prove an Onsager-type singularity theorem which shows that dissipative anomalies can appear for strong limits of compressible Navier-Stokes solutions only if the limiting Euler solutions have low regularity (sub-K41), of the type observed empirically in compressible turbulence. Surprisingly, anomalous energy dissipation for compressible Navier-Stokes solutions can be due not only to nonlinear energy cascade, but can also be due to a "pressure-work defect.” The latter mechanism is shown to occur for stationary, planar shocks with an ideal-gas equation of state. Furthermore, we find that there can also be anomalous entropy production for limiting Euler solutions, again illustrated by shocks, and due to both inverse cascade of entropy and pressure-work defect. Our proof of the Onsager singularity theorem for compressible fluids is based essentially on the entropy. These results extend as well to relativistic fluid turbulence, with some additional interesting twists. (The talk is based entirely on joint work with Gregory Eyink).

Presentation (PDF File)

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