Why can't there be a general theory of nonlinear PDE?

Question

Lawrence Evans wrote in discussing the work of Lions fils that

there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probabilistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods.

To me the second part of Evans' quote does not necessarily imply the first. So my question is: why can't there be a core theory of nonlinear PDE?

More specifically it is not clear to me is why there cannot be a mechanical procedure (I am reminded here by [very] loose analogy of the Risch algorithm) for producing estimates or good numerical schemes or algorithmically determining existence and uniqueness results for "most" PDE. (Perhaps the h-principle has something to say about a general theory of nonlinear PDE, but I don't understand it.)

I realize this question is more vague than typically considered appropriate for MO, so I have made it CW in the hope that it will be speedily improved. Given the paucity of PDE questions on MO I would like to think that this can be forgiven in the meantime.

Answer 1

I find Tim Gowers' "two cultures" distinction to be relevant here. PDE does not have a general theory, but it does have a general set of principles and methods (e.g. continuity arguments, energy arguments, variational principles, etc.).

Sergiu Klainerman's "PDE as a unified subject" discusses this topic fairly exhaustively.

Any given system of PDE tends to have a combination of ingredients interacting with each other, such as dispersion, dissipation, ellipticity, nonlinearity, transport, surface tension, incompressibility, etc. Each one of these phenomena has a very different character. Often the main goal in analysing such a PDE is to see which of the phenomena "dominates", as this tends to determine the qualitative behaviour (e.g. blowup versus regularity, stability versus instability, integrability versus chaos, etc.) But the sheer number of ways one could combine all these different phenomena together seems to preclude any simple theory to describe it all. This is in contrast with the more rigid structures one sees in the more algebraic sides of mathematics, where there is so much symmetry in place that the range of possible behaviour is much more limited. (The theory of completely integrable systems is perhaps the one place where something analogous occurs in PDE, but the completely integrable systems are a very, very small subset of the set of all possible PDE.)

p.s. The remark Qiaochu was referring to was Remark 16 of this blog post.

Answer 2

Further to my comment above, on the theorem of Pour-el and Richards: it originally appeared in Advances in Math. 39 (1981) 215-239, entitled "The wave equation with computable initial data such that its unique solution is not computable." I think it is fair to say that they get the wave to simulate a universal Turing machine, albeit with very complicated encoding. However, this may all be irrelevant to explaining why "nonlinear PDE are hard" because the wave equation is linear!

Answer 3

I agree with Craig Evans, but maybe it's too strong to say "never" and "impossible". Still, to date there is nothing even close to a unified approach or theory for nonlinear PDE's. And to me this is not surprising. To elaborate on what Evans says, the most interesting PDE's are those that arise from some application in another area of mathematics, science, or even outside science. In almost every case, the best way to understand and solve the PDE arises from the application itself and how it dictates the specific structure of the PDE.

So if a PDE arises from, say, probability, it is not surprising that probabilistic approximations are often very useful, but, say, water wave approximations often are not.

On other hand, if a PDE arises from the study of water waves, it is not surprising that oscillatory approximations (like Fourier series and transforms) are often very useful but probabilistic ones are often not.

Many PDE's in many applications arise from studying the extrema or stationary points of an energy functional and can therefore be studied using techniques arising from calculus of variations. But, not surprisingly, PDE's that are not associated with an energy functional are not easily studied this way.

Unlike other areas of mathematics, PDE's, as well as the techniques for studying and solving them, are much more tightly linked to their applications.

There have been efforts to study linear and nonlinear PDE's more abstractly, but the payoff so far has been rather limited.

Answer 4

To elaborate on Steve Huntsman's comment, I remember reading the following on Terence Tao's blog: there exist PDE that can simulate Newtonian mechanics, and using such a PDE and the correct initial conditions it is possible, in principle, to simulate an arbitrary analog Turing machine. So a general-purpose algorithm to determine even the qualitative behavior of an arbitrary PDE cannot exist because such an algorithm could be used to solve the halting problem.

Answer 5

Some more random thoughts:

The closest thing I've ever seen to a "general theory of nonlinear PDE's" is Gromov's book, Partial Differential Relations. He does many things in there that I don't understand, but one application that he applied his theory to is isometric embeddings of Riemannian manifolds into Euclidean space or other higher dimensional Riemannian manifolds (the problem made famous by Nash).

Moreover, in a paper by Bryant, Griffiths, and me (but in a section written by the other two and not me), it is shown that in some sense, the linearized PDE corresponding to the isometric embedding of an $n$-dimensional Riemannian manifold into $n(n+1)/2$-dimensional Euclidean space looks like a generic $n$-by-$n$ system of first order linear PDE's. I'm not aware of any other place where a "generic" system of PDE's arises naturally.

The results in this paper inspired some efforts by Jonathan Goodman and me (unpublished) as well as Nakamura and Maeda (TAMS 313 (1989) 1-51) to extend Hormander's theory of linear PDE's (at least those of real principal type) to nonlinear PDE's. (It should be noted that much more interesting work in this direction was done for the 2-dimensional case, starting with the Ph.D. thesis of C. S. Lin)

But maybe you really meant "the general theory of nonlinear PDE's that are elliptic, hyperbolic, or parabolic" and not really the all encompassing "general theory of nonlinear PDE's"? There's far too much junk in the latter.

Answer 6

I will simply quote Heisenberg. (This an approximative quote from memory.)

One can say almost everything about nothing, and almost nothing about everything.

Answer 7

I think there is something you can call a general theory of PDEs. It started already long time ago with Meray, Riquier, Janet, Elie Cartan. There is an important survey article by Donald Spencer: Overdetermined systems of linear partial differential equations , Bull. Amer. Math. Soc. 75 (1969), 179-239. see also the recent book by Seiler: Involution:The Formal Theory of Differential Equations and its Applications in Computer Algebra, springer, 2010. This book contains lots of references to this topic.

It is a bit strange why this line of research is not very well known.

Answer 8

this is a comment to Deane Yang, but apparently it was too long so here is a separate answer. My background is in numerical solution of PDEs

1) while I know about this, it is not at all well-known by people who numerically solve PDEs.

2) this is not true. Most computations are systems of PDEs. I think most computations are done with systems where there are no actual theory, i.e. existence and uniqueness results. Think about Navier-Stokes. Many systems are NS coupled with for example convection diffusion type systems (small amounts of material in the flow etc). Then there are liquid crystals, Maxwell, elasticity, flow coupled with elasticity etc. Of course when the computers were slower one had to simplify to get a scalar equation and then hope that it gives something reasonable.

Of course Cauchy-Kovalevskaia as such is irrelevant because one wants the solutions in Sobolev spaces. But the whole formal theory started as a generalization of CK.

3) this is not true. For example there are systems which are not elliptic initially but whose involutive forms are elliptic. This gives a priori regularity results and existence results.

Also one could argue that the word "determined" (and over/underdetermined) can't be defined in general without formal theory.

Answer 9

In my limited experience, the furthest you can carry a general theory of PDEs, assuming only smoothness of the PDEs for example, is to describe the characteristic variety and its integrability, or to determine whether the equations are formally integrable (in the sense of the Cartan-Kaehler theorem). Already you find that every real algebraic variety is the characteristic variety of some system of PDE. Even when the characteristic variety is very elementary (a sphere, for example), we know very little about the PDE (it is hyperbolic, but we don't have a complete theory of boundary value problems, initial value problems, long term existence, uniqueness). So I think that a general theory of PDE would have to be much more difficult than real algebraic geometry, which already has elementary problems that seem to be very difficult.

Answer 10

Yes there is! This needs to be highlighted so that nobody misses the memo:

The theory of Jet Bundles = The general theory of differential equations.

It is the mathematical formalism that provides a general and universal framework for both ordinary and partial differential equations.

A standard application of Jet Bundles is that of finding the symmetries of a set of equations that may include ordinary algebraic equations (i.e. order 0 differential equations), ordinary and/or partial differential equations within it. Here's an example: find the Symmetry Group Of Newton's First Law (It's SL(5)).

Now, here's a question that expands on the issue: is there a general framework for differential equations - including non-linear ones - for non-commutative algebras and/or generalized functions? For instance, is there a non-commutative or distributional version of Jet Bundle theory? There is such a thing as Non-Commutative Geometry, though I don't know if that's the place where non-commutative differential equations live.