Timeline for Is there a solution of a first order nonlinear PDE?
Current License: CC BY-SA 4.0
14 events
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| Jul 24, 2020 at 15:26 | comment | added | liding | Thank you very much! | |
| Jul 24, 2020 at 10:18 | comment | added | Robert Bryant | This isn't a general first order scalar PDE, but of a particular sort: If $M$ is a Riemannian manifold with metric $a$, $b$ is a vector field on $M$, and $c$ and $f$ are functions on $M$, the equation becomes $$|\nabla u|_a^2 + 2b\,\cdot\nabla u+c\,u - f =0.$$ (In the flat case, the inequalities on $a$ ensure that the metric is complete.) Writing it as $$|\nabla u+b|_a^2 + c\,u - (f+|b|^2_a) = 0,$$ makes it clearer when solutions exist in a neighborhood of a point. For example $c(x)\not=0$ or $f(x)+ |b(x)|^2_a>0$ would guarantee local solutions near $x$, by the method of characteristics. | |
| Jul 24, 2020 at 6:30 | comment | added | Andrew | @liding The eikonal equation $|\nabla u|=f$ is a special case. Lot is known about it. | |
| Jul 24, 2020 at 6:08 | history | edited | liding | CC BY-SA 4.0 |
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| Jul 22, 2020 at 16:22 | comment | added | liding | Could you give me some references | |
| Jul 22, 2020 at 14:48 | history | edited | liding | CC BY-SA 4.0 |
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| Jul 22, 2020 at 14:43 | comment | added | liding | I have a PDE similar with this problem on closed manifold. I just want to find a solution. Both the sufficient conditions and the local solution will help me. | |
| Jul 22, 2020 at 9:18 | comment | added | Robert Bryant | There's not always a solution. Just look at $(u')^2 = -1$ when $n=1$. Are you asking for sufficient conditions that a solution exist? Would existence of local solutions suffice for what you want? Perhaps you can make your question more precise. | |
| Jul 22, 2020 at 7:25 | history | edited | liding | CC BY-SA 4.0 |
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| Jul 22, 2020 at 7:24 | comment | added | liding | Yes ,it is the partial derivative with respet to $x_{i}$. Thank you and D. Tampieri for pointing out it. | |
| S Jul 22, 2020 at 7:21 | history | suggested | Daniele Tampieri | CC BY-SA 4.0 |
Minor typo correction
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| Jul 22, 2020 at 6:56 | review | Suggested edits | |||
| S Jul 22, 2020 at 7:21 | |||||
| Jul 22, 2020 at 6:51 | comment | added | David Roberts♦ | Is $u_i$ the partial derivative with respect to $x_i$, or something else? | |
| Jul 22, 2020 at 5:20 | history | asked | liding | CC BY-SA 4.0 |