Timeline for Definitions of weak solutions for quasilinear wave equations
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 19 at 2:13 | vote | accept | lsb | ||
| Apr 17 at 11:06 | answer | added | Willie Wong | timeline score: 2 | |
| Apr 17 at 9:24 | comment | added | Willie Wong | Okay, now that I understand where you are coming from, I think you are mostly correct and I can provide an answer. | |
| Apr 17 at 8:08 | comment | added | Willie Wong | I think I misunderstood your question originally; so I deleted a bunch of my comments. I'll think a bit more and come back. | |
| Apr 17 at 7:30 | comment | added | lsb | @WillieWong Sorry for the further query. As your comment, but how to define the RH condition for above quasilinear wave equations? I only saw RH conditions for the 1st order hyperbolic system in the literatures. Thus should we rewrite this quasilinear wave equation to a 1st order hyperbolic system and then use the wellknown RH condition? In that case, does it also mean we know how to define the weak solution too? | |
| Apr 17 at 7:19 | comment | added | lsb | @WillieWong Thanks Willie. I might catch something. | |
| Apr 17 at 7:15 | comment | added | lsb | @WillieWong In my understanding, I thought the RH condition is equivalent to a weak solution, so if we do not have a definition of weak solution, how to give the RH condition? I also notice references by Majda from your article (arxiv.org/pdf/1407.6276.pdf). He consider the first order hyperbolic system, and he still use the weak solution and give the Rankine-Hugoniot condition. Hence, I am thinking if we can transform above wave equation to a first system to use Majda's approach? | |
| Apr 17 at 7:06 | comment | added | lsb | @WillieWong Thanks for the comment, but this comment may cause me two further confusions: 1. If you consider a free boundary problem, how should we give the free boundary condition? Since if we have a weak solution, it is naturally equivalent to the Rankine-Hugoniot condition, otherwise, what is a natural and reasonable boundary condition? | |
| Apr 17 at 6:08 | comment | added | Willie Wong | I don't think modern approaches go via the weak solution route. The weak solution theory developed for a single spatial dimension uses $L^1$ spaces and is well-known to be not compatible with higher dimensional wave equations; this is related to the fact that method of characteristics fail. Instead of directly generalizing the class of solutions considered, current approaches study instead a free boundary problem (the boundary is the shock front). The solution is classical away from the front. | |
| Apr 17 at 2:35 | history | edited | lsb | CC BY-SA 4.0 |
added 83 characters in body
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| Apr 17 at 2:30 | history | asked | lsb | CC BY-SA 4.0 |