Question

I am currently trying to understand the notion of criticality (as discussed, e.g., in Terence Tao's book on nonlinear dispersive equations) from a physical viewpoint. That's why i'm interested in the question which PDE from physics (apart from the Navier-Stokes equations) and geometry are supercritical with respect to some symmetry and all (known) controlled quantities.

Answer 1

Generally speaking, supercriticality occurs when the dimension and/or the nonlinearity exponent is sufficiently large.

Sigma field models such as the harmonic map, wave map, or Schrodinger map equations become supercritical in three and higher spatial dimensions. (The critical two-dimensional case is probably the most interesting.)

Einstein's equations of general relativity also becomes supercritical in three and higher spatial dimensions. The closely related Ricci flow used to be supercritical in three and higher dimensions, until Perelman discovered some new scale-invariant controlled quantities; now I would classify it as critical in three dimensions at least, and possibly in higher dimensions (though it is not as clear there whether Perelman's quantities are coercive enough to fully control the dynamics at small scales). (In general, elliptic and parabolic equations can defy to some extent the criticality classification arising from dimensional analysis, due to powerful monotonicity formulae such as those arising from the maximum principle.)

Yang-Mills equations are supercritical in five and higher spatial dimensions, and similarly for related equations such as the Maxwell-Klein-Gordon equations. (Yang-Mills theory becomes particularly interesting in the critical four-dimensional case, what with its instantons, self-dual and anti-self-dual solutions, etc.)

For pure power nonlinearity interactions (with a term of the form $|\phi|^p$ in the Hamiltonian), one typically has supercriticality once the exponent p becomes large enough, although the precise threshold of p depends on the dimension and on the precise model. For example, $|\phi|^4$ models generally become supercritical in five and higher spatial dimensions. Milder nonlinearities, such as Hartree-type nonlinearities, tend to be less supercritical than power nonlinearities.

Navier-Stokes is supercritical in three and higher dimensions. One can certainly perform the relevant dimensional analysis on other fluid equations (e.g. quasi-geostrophic), but I don't recall the exact numerology off-hand. But in the absence of viscosity (e.g. for the Euler equations), there is now a two-parameter family of scaling invariances, and there is not really a well-defined notion of criticality, subcriticality, or supercriticality in this case.

For systems of coupled equations (e.g. Zakharov type models) for which there is no natural scaling, it becomes more difficult (and perhaps even impossible) to cleanly make the division into subcritical, critical, and supercritical equations; the distinction is most useful for simplified model equations.

Answer 2

I'm not an expert on GR at all, but looking at the cosmos I guess that solutions to Einstein's equations do not exhibit the self-randomization that lies at the heart of fluid turbulence. This might suggest that the problem of showing the existence of smooth solutions of NSE is related either to supercriticality or to turbulence only superficially. I tend to the latter opinion. I've been reading literature on turbulence research for some time now and I'm getting the impression that it's not very natural to expect a deep link between turbulence and the Millenium problem. We know that NSE in two dimensions possess smooth solutions. But the proof of existence of these solutions adds no insight to two dimensional turbulence. I guess that research in fluid turbulence has the potential to open the gate to a new world of PDE theory and infinite dimensional dynamical systems but i doubt that this is closely related to the existence of smooth solutions. Of course, all this is extremely vague and I might well be completely wrong.