  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.
