I'm sorry if this is an inappropriate forum to ask this question on, for I fear it is pretty undergraduate-level one :) I was contemplating on the study of non-linear PDEs. Is it possible to reduce a non-linear PDE on $\mathbb{R}^n$ to a distribution or a 'good' PDE on a smooth manifold? It seems to me like a natural step, but I don't know anything about it yet :(
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5$\begingroup$ too vague a 'question'. $\endgroup$– Denis SerreCommented Sep 22, 2010 at 5:23
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$\begingroup$ I agree with Dennis Serre. What do you mean by 'good'? Do you have a simple example or heuristic? $\endgroup$– hceCommented Sep 22, 2010 at 7:21
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6$\begingroup$ I take 'good' to mean able to solve, intepret, or extract qualitative behaviour due to the reduction/transformation. I think the question makes sense and that some vagueness correlates well with how ad hoc the study of PDEs is. Perhaps writing a simple PDE on the sphere and then using stereographic projections to put it on R^2 could help to reverse-engineer an illustrative example. $\endgroup$– Q.Q.J.Commented Sep 22, 2010 at 7:46
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2$\begingroup$ You may want to look up the formalism of exterior differential systems which has a bit of the flavor you are describing. In particular, there one can use the equations of a system of PDEs to formally cut out a submanifold of a jet bundle and then solutions to the PDE are in correspondence with subsets where certain differential forms vanish and others are non-vanishing. See e.g. "Cartan for Beginners" by Ivey and Landsberg. $\endgroup$– j.c.Commented Sep 22, 2010 at 13:40
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1$\begingroup$ Oh, forgot to look at the tag. In that case, as jc pointed out, the thing to look for are exterior differential systems and jets. Another reference is the book "Symmetries and Conservation laws for Differential Equations of Mathematical Physics" from Krasilshchik Vinogradov et. al.. Despite the specialized sounding title it's an introduction to this geometric approach. $\endgroup$– Michael BächtoldCommented Sep 22, 2010 at 15:25
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1 Answer
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The question is vague (Denis is right) but it does make some sense. When looking at explicit solutions of nonlinear PDEs one frequently encounters singularities that remind of projections from some higher dimensional manifold (e.g. in Burger's equation). But I think the OP does not have a clear picture. Locally, working 'on' a manifold instead than on $R^n$ is essentially like changing coordinates in the equation. So usually you do not resolve nonlinearities, you just modify the coefficients of the equation.
On the other hand, if you consider equations with values into a manifold, then sometimes a very difficult nonlinear term reveals to be a very simple geometric object, 'linear' in some sense. This does not happen by chance; your equation must have some geometric meaning. Best examples are harmonic maps (elliptic) and wave maps (hyperbolic).
Some years ago, Serge Alinhac tried to develop a whole theory of 'geometric blow up': singularities in the solution which can be resolved by choosing an appropriate coordinate sistem in the target space. But I guess the results where not too satisfying since he abandoned the effort.
answered Sep 22, 2010 at 9:32
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1$\begingroup$ @Piero: do you have a reference for the geometric blow up that Alinhac tried? It sounds interesting. $\endgroup$ Commented Sep 22, 2010 at 14:33
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$\begingroup$ There is a <a href = "openlibrary.org/books/OL1271499M… "> book </a> where he collected notes of Orsay courses (~94?) on the subject. Quite interesting $\endgroup$ Commented Sep 23, 2010 at 9:14