Onsager's "Ideal Turbulence" Theory

Abstract

Lars Onsager in 1945-1949 made an exact analysis of the high Reynolds-number limit for individual turbulent flow realizations modeled by incompressible Navier-Stokes equations, motivated by experimental observations that dissipation of kinetic energy does not vanish. I review here developments spurred by his key idea, that such flows are well-described by distributional or "weak" solutions of ideal Euler equations. 1/3 H\"older singularities of the velocity field were predicted by Onsager and since observed. His theory describes turbulent energy cascade without probabilistic assumptions and yields a local, deterministic version of the Kolmogorov 4/5th law. The approach is closely related to renormalization group methods in physics and envisages "conservation-law anomalies", as discovered later in quantum field theory. There are also deep connections with Large-Eddy Simulation modeling. More recently, dissipative Euler solutions of the type conjectured by Onsager have been constructed and his 1/3 H\"older singularity proved to be the sharp threshold for anomalous dissipation. This progress has been achieved by an unexpected connection with work of John Nash on isometric embeddings of low regularity or "convex integration" techniques. The dissipative Euler solutions yielded by this method are wildly non-unique for fixed initial data, suggesting "spontaneously stochastic" behavior of high-Reynolds number solutions. I focus in particular on applications to wall-bounded turbulence, leading to novel concepts of spatial cascades of momentum, energy and vorticity to or from the wall as deterministic, space-time local phenomena. This theory thus makes testable predictions and offers new perspectives on Large-Eddy Simulation in presence of solid walls.