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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.

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Are there any Markov or Novikov type theorems for PDEs ? i.e. presumably you could encode algorithmically unsolvable problems into the language of PDEs. Meaning, knowledge of some aspect of the solution (bounded orbit, say) is equivalent to knowing the solution to an algorithmically unsolvable problem? If there were such theorems that would partially address your question. –  Ryan Budney Feb 14 '10 at 23:14
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Perhaps the kind of negative result you are looking for is the theorem of Pour-el and Richards that the 3-dimensional wave equation has non-computable solutions with computable initial conditions. This is in their book Computability in Analysis and Physics (Springer-Verlag 1989). –  John Stillwell Feb 14 '10 at 23:43
    
@Ryan, John--Good point! (One or both of) You should put this in an answer...I seem to recall hearing something once along the lines of PDEs being Turing-universal. But perhaps there can be a general theory of PDE that correspond to a restricted model of computation? –  Steve Huntsman Feb 15 '10 at 1:10
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When I teach basic differential equations, I stress analogies with algebraic equations. While this is probably more simple-minded than you were looking for, I point out (without attempting a thorough justification) that although there is a good theory of linear (algebraic equaions) a general theory to solve all algebraic equations, no matter how irregular, is hopelessly out of reach. And we have no right to expect better of differential equations. –  Mark Meckes Mar 23 '10 at 16:31
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This also brings to mind the preface (books.google.com/…) from "Lectures on Partial Differential Equations" by Arnol'd. Unfortunately the google books version cuts out after the first page, and I can't find another English version online. You can find a Russian version by googling for "Лекции об уравнениях с частными производными". –  Josh Guffin Sep 29 '10 at 18:26
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9 Answers

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.

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I wonder: can one not model Turing machines using ODEs? –  Mariano Suárez-Alvarez Feb 15 '10 at 2:15
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And even the completely integrable systems are full of surprises, such as the Camasso–Holm equation, where the solution concept needs some tweaking in order to make the Cauchy problem well posed. –  Harald Hanche-Olsen Feb 15 '10 at 2:20
    
@Mariano: yes, as covered in your subsequent question: mathoverflow.net/questions/15309 –  Steve Huntsman Feb 15 '10 at 4:26
    
Leave it to Terry Tao to give the most knowledgable and succinct response to a deep question.His grasp of the Big Picture and relevant publications in any field never ceases to amaze me. –  Andrew L Jun 4 '10 at 21:13
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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!

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Yes, I would say that there is a general theory of linear PDE, and Hörmander pretty well captures the basics. –  Steve Huntsman Feb 15 '10 at 3:00
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Yes, there is a general theory of linear PDE developed largely by Hormander, but of what use is it? In some sense, the space of all possible linear PDE's can be viewed as a singular algebraic variety, where Hormander's theory applies only to generic (smooth) points and the most interesting and heavily studied PDE's all lie in a lower-dimensional subvariety and mostly in the singular set of the variety. –  Deane Yang Feb 15 '10 at 3:17
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Also, even though you can't solve the halting problem for Turing machines, the existence, uniqueness, and computability (by definition!) of solutions to the Turing machine “equations of motion” are all utterly trivial. For PDEs, nothing could be farther from the truth. Similarly for ODEs: The local theory is easy, it's long term and global behaviour that is difficult. But for PDEs, even the local theory is fiendishly difficult. (Except for the Cauchy-Kowalevskaja theorem, which despite (or because of?) its generality also turns out to be of rather limited use.) –  Harald Hanche-Olsen Feb 15 '10 at 3:31
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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.

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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.

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As far as my very limited understanding goes, the h-principle is not really a "general theory of nonlinear PDE's" but mostly applies to underdetermined systems, which happen to arise a lot in geometric applications, but not as much in physics. –  Otis Chodosh Jan 11 '12 at 3:54
    
Yes, Gromov's study of PDE's is pretty much limited to underdetermined systems and therefore is definitely not a study of general PDE's. But it applies to general underdetermined PDE's, and that's probably the broadest class of PDE's that anyone has been able to study using a unified approach. –  Deane Yang Jan 11 '12 at 8:39
    
Here is a 7-page review of Partial Differential Relations by Dusa McDuff: projecteuclid.org/DPubS/Repository/1.0/… –  Tom LaGatta Mar 20 '13 at 10:44
    
Tom, thanks. I've certainly seen that when it first came out. –  Deane Yang Mar 21 '13 at 2:41
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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.

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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.

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In response to "It is a bit strange why this line of research is not very well known": 1) Actually, this stuff has become much better known through the work and books by Bryant, Chern, Goldschmidt, Griffiths, Ivey, and Landsberg. 2) Most PDE's that arise from other areas of mathematics and sciences are either scalar or determined systems. For such PDE's, the formal theory tells you nothing more than what the Cauchy-Kovalevski theorem says. 3) The formal theory tells you nothing about the global behavior and regularity of solutions to PDE's. –  Deane Yang Jun 4 '10 at 18:47
    
@Deane. Your comment 2) is irrelevant for several reasons. a) Cauchy-Kovalevskaia theorem tells you nothing about the Cauchy problem for the heat equation, Navier-Stokes system or Schrödinger equation, because the order with respect to time ($=1$) is smaller than the total order ($=2$). b) Real problems are posed in domains with boundaries, and the boundary conditions can be non-homogeneous. You may need a very much elaborated theory to prove the solvability. Hyperbolic initial-boundary-value problems are notoriously difficult (see the book by S. Benzoni-Gavage and myself); C.-K. is useless. –  Denis Serre Nov 18 '10 at 7:51
    
Denis, your statements are consistent with and provide some details that underly mine. –  Deane Yang Jan 20 '11 at 4:27
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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.

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Here are my reactions: 1) Is the formal theory useful for numerical solutions? Could you provide references for this? 2) There are certainly systems consisting of an evolution equation that is coupled with a constraint or gauge condition. Navier-Stokes is like this. The formal theory provides no new insights for these systems, either. 3) What example do you have in mind? I know this statement as an abstract theorem, but I have never seen it used anywhere. –  Deane Yang Jun 4 '10 at 20:26
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I will simply quote Heisenberg. (This an approximative quote from memory.)

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

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But why is the study of PDE's "everything"? In comparison to, say, the study of polynomials? –  Deane Yang Jan 11 '12 at 8:37
    
Of course, the study of PDEs is not everything, but the point of Heisenberg's quote (and he was referring specifically to nonlinear pde-s) is within the pde Universe, the statement that something is nonlinear carries zero information. PS The study of polynomials can be viewed as a subclass of the study of PDE's (think characteristic polynomials of constant coefficients o.d.e.'s) More generally the theory of D-modules suggests that substantial chunk of algebraic geometry is closely connected to PDE's. –  Liviu Nicolaescu Jan 11 '12 at 11:28
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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.

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