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.
[Math] Which PDE from physics (and geometry) are supercritical
ap.analysis-of-pdes
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As alluded-to by Qiaochu Y. above, and as I can personally attest, PDE arise in the modern theory of automorphic forms. Superficially/historically, this might be viewed as a formal generalization of "holomorphic" to "eigenfunction for Laplace-Beltrami operator". Indeed, already c. 1947, Maass showed that real quadratic fields' grossencharacter L-functions arose as Mellin transforms of "waveforms", Laplace-Beltrami eigenfunctions on $\Gamma\backslash H$, a complementary result to his advisor Hecke's result that $L$-functions for complex quadratic extensions of $\mathbb Q$ arose from holomorphic modular forms.
The spectral theory of automorphic forms, from Avakumovic, Roelcke, and Selberg c. 1956, in effect decomposes $L^2(\Gamma\backslash H)$ with respect to the invariant Laplacian, descended from the Casimir operator on the group $SL_2(\mathbb R)$, which (anticipating theorems of Harish-Chandra) almost exactly corresponds to decomposition into irreducible unitary representations.
The Selberg trace formula, and Langlands' and Arthur's, as well as Jacquet's "relative" trace formula, do afford an interpretation as spectral decompositions of various integral operators, rather than differential operators. Nevertheless, or "however", some aspects of the situation that are clumsy, because of their "extreme" features, but interesting for applications for the same reason, from that viewpoint are amenable to thinking about solutions of (invariant) inhomogeneous PDEs with distributional "targets". A typical scenario is a "Helmholtz" equation (a wave equation Fourier-transformed in the time parameter), $(\Delta-\lambda)u=f$. Among other cases of interest, the case that $f$ is an (automorphic) delta is very useful in various number-theoretic applications, such as proving "subconvex" bounds: Anton Good sketched this application already in 1983 (and Diaconu and I treated $GL_2$ over number fields recently... implicitly using this idea, although reference to classical special functions gave a shorter argument for the official version).
Philosophizing a bit, such experiences, and continuing ones of a related sort, indicate to me that geometrically meaningful, that is, group-invariant, "PDE" are a natural/obvious extension of "calculus"... so that, in particular, their natural solutions in Sobolev spaces (etc) are "natural objects", whether or not they are classical special functions, or entirely elementary.
(One can't help but note that there is an understandable, if unfortunate, human tendency to declare and understand "turf", so that one chooses one's own, and stays away from others'. Similarly, "experts" on subject X do not favor outsiders' appropriating bits of it "for applications", as though anything other than a life-long dedication could penetrate the mysteries... One may read about medieval European "guilds" and their protection of their "secrets".)
As a methodological philosophizing: my own experience tells me that means of description are useful. That is, structural, meaningful characterization of objects is good. Saying that something is a solution of a natural (group-invariant?...) PDE is a strong, meaningful constraint. Ergo, helpful/good.
The small rant at the end: the usual style of seemingly-turf-respecting narrowness is not so good for genuine progress, nor even for individual understanding.
A very active field of research (and to my understading, may fall into the "fundamental" category) is Domain Decomposition methods (DDM), which can be understood in the geometrical numerical and computational sense. In this last, parallel algorithms are being explored. Although many of these methods are based on Lagrange multipliers, efforts are also made to make a sensible domain decomposition without using them, through indirect methods, like Green's functions, from which some collocation methods can be derived (see "General Theory of Domain Decomposition: Indirect Methods Ismael Herrera, Robert Yates, Martin Diaz", Numerical Methods of Partial Differential Equations, Wiley). In this same paper are mentioned the Steklov-Poincare operators, which I believe is a line of research on its own right. And come to think of it, collocation methods are also a line of research.
Your related question: "[...]fundamentally new types of functional spaces[...]" You don't mention which functional spaces you are aware of, but I can mention Sobolev spaces, which support weak derivatives, and these in turn can be used in problems involving "jumps" or some type of discontinuity (for example, in combustion/explosion problems, see in this instance "physics of shock waves and High-Temperature Hydrodynamic phenomena" by Ya. B. Zeld'dovich and Yu. P. Raizer"). See also Godlewski and Raviart (1996), "Numerical approximation of hyperbolic systems of conservation laws", Applied Mathematical Sciences 118 (Springer, New York). Sobolev spaces have been used also for DDM, for example in the same paper by Herrera, Yates and Diaz.
There is this book "Navier-Stokes Equations and Turbulence" edited by C. Foias, which devotes some pages to the Banach-Tarski paradox. Maybe that famous paradox (a theorem, in fact) may provide some new avenues of research into some types of PDE's.
Best Answer
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.