Timeline for Why can't there be a general theory of nonlinear PDE?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 29, 2015 at 19:18 | review | Late answers | |||
| Sep 29, 2015 at 19:53 | |||||
| Jan 20, 2011 at 4:27 | comment | added | Deane Yang | Denis, your statements are consistent with and provide some details that underly mine. | |
| Nov 18, 2010 at 7:51 | comment | added | Denis Serre | @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. | |
| Jun 4, 2010 at 18:47 | comment | added | Deane Yang | 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. | |
| Jun 4, 2010 at 18:29 | history | answered | jukka tuomela | CC BY-SA 2.5 |