Timeline for Non-linear, hyperbolic, 2nd order system of PDEs
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Oct 24, 2021 at 17:00 | history | bounty ended | CommunityBot | ||
| S Oct 24, 2021 at 17:00 | history | notice removed | CommunityBot | ||
| S Oct 16, 2021 at 15:32 | history | bounty started | Daniel Castro | ||
| S Oct 16, 2021 at 15:32 | history | notice added | Daniel Castro | Draw attention | |
| Oct 11, 2021 at 18:29 | comment | added | Deane Yang | What is Gouraat data? | |
| Oct 11, 2021 at 17:55 | history | edited | Daniel Castro | CC BY-SA 4.0 |
cross-post
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| Oct 11, 2021 at 7:56 | comment | added | Daniele Tampieri | Daniel, just for the sake of correctness, It would be advisable to add a remark to point out that you have already asked the same question several days ago and no one has answered. | |
| Oct 11, 2021 at 1:59 | comment | added | Willie Wong | Ignoring convergence issues, just thinking about the solvability of the power series expansion near the origin, it seems to me you need to also prescribe at least $\partial_y \alpha (x,0)$ and $\partial_x \beta(0,y)$ to have a chance of guaranteeing that the solution you write down is unique. | |
| Oct 11, 2021 at 1:51 | comment | added | Willie Wong | Are you sure you are giving enough initial data? For a system like this, usually local existence and uniqueness is independent of lower order perturbations. So consider the simplified equation where $K = 0$. Suppose $\alpha(x,0) = a_0$ and $\beta(0,y) = b_0$, then any pair of linear functions of the form $\alpha(x,y) = a_0 + \gamma y$ and $\beta(x,y) = \beta_0 + \delta x$ would solve the system. I expect similar counter examples can be found after restoring $K > 0$. | |
| Oct 11, 2021 at 1:44 | comment | added | Deane Yang | It was just a hopeful but not really serious question. The noncharacteristic initial value problem is well-posed, and there exists a unique solution in a neighborhood of the line. It also seems possible that if the initial data is sufficiently small, a globl solution exists. | |
| Oct 10, 2021 at 22:25 | comment | added | Daniel Castro | @DeaneYang Thank you. Non-characteristic lines are not our primary focus, but any suggestions even in that case would be helpful. | |
| Oct 10, 2021 at 22:20 | comment | added | Deane Yang | Any chance you’d be interested in setting the initial data along the line $y=x$ or any other line but the axes? | |
| Oct 10, 2021 at 21:31 | history | asked | Daniel Castro | CC BY-SA 4.0 |