Timeline for Why can't there be a general theory of nonlinear PDE?
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| Feb 15, 2010 at 3:31 | comment | added | Harald Hanche-Olsen | 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.) | |
| Feb 15, 2010 at 3:17 | comment | added | Deane Yang | 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. | |
| Feb 15, 2010 at 3:00 | comment | added | Steve Huntsman | Yes, I would say that there is a general theory of linear PDE, and Hörmander pretty well captures the basics. | |
| Feb 15, 2010 at 2:28 | history | answered | John Stillwell | CC BY-SA 2.5 |