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.


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.

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    $\begingroup$ I was going to point to the dispersive wiki, but I think you pretty much covered all the examples mentioned there. $\endgroup$ – Willie Wong Apr 4 '12 at 12:41

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.

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    $\begingroup$ The reason the cosmos looks non-turbulent is because the initial data is small. If we were in a universe filled with black holes, neutron stars, and wormholes, we would see a much more turbulent experience. (Conversely, even in three dimensions, small perturbations of a stationary viscous fluid do not exhibit turbulence, as the dissipation dominates the nonlinear behaviour.) $\endgroup$ – Terry Tao Apr 4 '12 at 16:15
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    $\begingroup$ In two dimensions, which is critical, global regularity is only barely provable, and as a consequence, comes with quite poor quantitative bounds (in particular, one can start having energy seep into exponentially small scales before dissipation finally kicks in). Regularity itself is only the first step towards further analysis of a PDE; it is an important goal, but not the final one. $\endgroup$ – Terry Tao Apr 4 '12 at 16:16
  • $\begingroup$ I guess I should have submitted my last contribution as a comment, not as an answer. Sorry for that. I'm new to this forum. I wasn't aware of the fact that the initial data of our cosmos is small, as you say. Thanks for pointing this out. $\endgroup$ – Daniel Lengeler Apr 4 '12 at 19:23
  • $\begingroup$ Of course, I agree that regularity is the first step, not the last. But I think that it is generally hoped that the solution of the Millenium problem might give a strong hint as to which direction to follow in turbulence. By the way, it might as well be vice versa, namely that an advance in turbulence is needed before the Millenium problem can be solved. The idea I had in mind when posing my question is the following. Is it possible to extract some qualitative dynamical features that are common for all supercritical physical systems, and hence might be directly linked to supercriticality? $\endgroup$ – Daniel Lengeler Apr 4 '12 at 21:56
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    $\begingroup$ To paraphrase Tolstoy: "Subcritical PDE are all alike; but each supercritical PDE is supercritical in its own way." Once the nonlinear effects dominate the dispersion or dissipation, it is quite hard to predict (especially at our current level of understanding) what happens to solutions. One feature all of these equations tend to have in common though is that solutions tend to be quite sensitive to the choice of initial data when studied over long times and/or large initial data, leading to instability, blowup, and/or rapid growth of various regularity norms. $\endgroup$ – Terry Tao Apr 4 '12 at 22:26

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