Timeline for Posing Cauchy data for the heat equation: $t=0$ a characteristic surface?
Current License: CC BY-SA 2.5
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| Oct 10, 2011 at 14:29 | comment | added | timur | @Dorian: From the point of view of the Cauchy-Kovalevskaya theorem, you need to specify the time derivative $u_t$ of the solution also as an initial datum at the initial surface. But this derivative is given by the heat equation itself, hence $u_t$ cannot be specified arbitrarily, which is the consistency condition you are talking about. Where is the corresponding non-uniqueness? It is when the initial datum grows fast at infinity. | |
| Sep 11, 2010 at 18:40 | comment | added | Deane Yang | Unfortunately, I don't know any specific reference for what you want. You might want to look in some more classical texts on PDE's. Or a book on "mathematical methods for physics". Also, there is a discussion of the heat equation on the half line here: en.wikipedia.org/wiki/Heat_equation | |
| Sep 11, 2010 at 18:22 | comment | added | Dorian | If the question is something very standard then alternatively I'd appreciate direction to a good source on this. Thanks! | |
| Sep 11, 2010 at 18:21 | comment | added | Dorian | I'm reading mostly from Evans and Folland and not much discussion is made regarding this issue. I've seen many times the discussion of consistency of boundary data and cauchy data for the wave equation. For the heat equation however I have yet to come across any remarks about consistency of boundary data with cauchy data. I'm sorry if my questions are vague but I'm having difficulties understanding these topics. Let me try to ask something more precise: Given the heat equation on $[0,\infty)$, can I pose any boundary data on $x=0$? What kind of consistency criterion do I need? | |
| Sep 11, 2010 at 16:28 | comment | added | Deane Yang | Dorjan, your questions are too vague for me to understand and answer. Certainly, if you are trying to solve an initial value problem on a space domain with boundary, you have to worry about the boundary data, just as you do for a hyperbolic PDE. For the classical heat and wave equations, this is explained pretty well in most PDE or mathematical physics texts. As for the equation $\partial_xu = 0$, it is definitely not parabolic, so there is no reason to expect the characteristic initial value problem to be well-posed. The behavior of a PDE depends in general on more than the top order terms. | |
| Sep 11, 2010 at 14:36 | comment | added | Deane Yang | Willie, according to my calculations I have had a 26 year head start on you. If one adjusts for that you appear to be well ahead of me. | |
| Sep 11, 2010 at 14:31 | comment | added | Dorian | Is there some sort of equivalent trouble for the heat equation? It seems like for parabolic equations I can pose ANY cauchy data I want as long as it is smooth enough and I get a solution. I just find this surprising considiner all of the other examples one encounters, there are noticable problems with consistency of cauchy data versus boundary data. Are parabolic equations free of this? | |
| Sep 11, 2010 at 14:30 | comment | added | Dorian | I just wanted to ask Thanks! I guess I'm surprised that we even can do it though. When focusing in on first order equations there are compatibility issues. For instance I can't pose Cauchy data for the equation $\partial_x u =0$ on $\mathbb{R}$ unless my Cauchy data is constant. For the wave equation I need to make sure that my Cauchy data does not conflict with my boundary data (for instance specifying both $u_t$ and $u_x$ on the boundary $x=0$ for the semi-infinite line problem would be a problem). | |
| Sep 11, 2010 at 14:25 | vote | accept | Dorian | ||
| Sep 11, 2010 at 12:42 | comment | added | Willie Wong | Oh Deane, how is it that you always manage to explain things so much better than I can? (Removed my more or less duplicate, but inferior answer.) | |
| Sep 11, 2010 at 11:58 | history | answered | Deane Yang | CC BY-SA 2.5 |