Timeline for Finite speed propagation by finite energy method
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 25, 2017 at 14:16 | history | edited | Willie Wong | CC BY-SA 3.0 |
added 3844 characters in body
|
| May 24, 2017 at 17:41 | vote | accept | J.Mayol | ||
| May 24, 2017 at 15:51 | comment | added | Willie Wong | (2) Probably $u_t^2 + 3 u^2 |\nabla u|^2$ is better. On domains where $u$ is bounded away from zero you get coercivity of the energy and by taking high enough derivatives everything goes through. // This is standard stuff that you can find in, e.g. Sogge's Nonlinear wave equations or Hormander's lecture notes that I mentioned in the previous comment. | |
| May 24, 2017 at 15:49 | comment | added | Willie Wong | (1) If $F(t,x, u, \nabla u, \nabla^2 u) = 0$ is a fully nonlinear perturbation of the wave equation, then taking the space-time gradient of the equation you get a bunch of quasilinear equations for the quantities $\nabla u$, by virtue of the chain and product rules of differentiation. This allows us to rewrite everything as a large system of quasilinear equations for the vector $(u, \nabla u)$. In Hormander's Lectures on Nonlinear Hyperbolic... this is Remark 4 at the end of Section 6.4 | |
| May 24, 2017 at 15:31 | comment | added | J.Mayol | Can you give a little more details for the part "This can be generalized to the case where the background equation has variable coefficients too, and so by freezing coefficients you get also a uniqueness theorem" I am intersted by how you do that. Do you consider the quantity $e(t)=u_t ^2/2 + u^2 |\nabla u|^2/2$ for the variable coefficients setting? | |
| May 24, 2017 at 15:00 | comment | added | J.Mayol | This is a well-documented answer! In fact, I already proved the finite speed propagation property far away from $u=0$ and I was looking for methods of proof for the finite speed propagation around $u=0$. By the way, to which argument of Hörmander do you refer to? | |
| May 24, 2017 at 14:30 | history | answered | Willie Wong | CC BY-SA 3.0 |