Timeline for Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?
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| Jun 3 at 4:03 | comment | added | monotone operator | @WillieWong, thank you very much! I will follow your idea to search this papers. Maybe my memory is wrong, last year I find a paper about evolution type Monge Ampere equation, but I forget the title of that paper, so I asked this question on the mathoverflow. Thank you again for your hints! Best wish! | |
| Jun 3 at 3:58 | comment | added | Willie Wong | @monotoneoperator Krylov has introduced the notion of a parabolic Monge-Ampere equation, but the equation is $ u_t \det(D^2 u) = -1$. If you Google "Parabolic Monge-Ampere" you should see some papers. I would be tempted to say that the correct hyperbolic analogue should be solving $\det(D^2 u) = -1$ assuming $D^2u$ has signature $(-+++\ldots)$. | |
| Dec 6, 2023 at 3:51 | comment | added | monotone operator | @DeaneYang thank you for your comments! I learn a lot from your comments! | |
| Dec 6, 2023 at 3:50 | comment | added | Deane Yang | That’s just not true. Most PDEs have no physical or geometric meaning. Only some of the ones you see in books and papers. There are many others you never see that are of no interest at all. The generic PDE is a terrible thing. | |
| Dec 6, 2023 at 3:38 | comment | added | monotone operator | Almost all PDE has its physical or geometrical meaning, my question just a connection in my mind. And, are there equations like $u_{t}=\det(D^{2}u)+\dots?$ | |
| Dec 6, 2023 at 3:34 | comment | added | monotone operator | Thank you,@DeaneYang, this equation maybe meaningless. When I meet hyperbolic mean curvature flow, I want to know whether there are equations like this form. | |
| Dec 5, 2023 at 18:31 | comment | added | Deane Yang | The question is why should we study this equation? The only hyperbolic PDEs studied extensively in geometric analysis are those in general relativity. These are already quite difficult to analyze, but people are motivated to do so by the physical significance of the equations. Why would we want to study this particular PDE? In general, nonlinear hyperbolic PDEs are much more difficult to study than noninear elliptic or parabolic PDEs. | |
| Dec 5, 2023 at 16:09 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
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| Dec 5, 2023 at 5:45 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Dec 4, 2023 at 3:48 | comment | added | monotone operator | @Denis Serre, Thank you for your correct, professor! | |
| Dec 3, 2023 at 13:04 | history | edited | Denis Serre | CC BY-SA 4.0 |
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| Dec 3, 2023 at 12:32 | history | edited | monotone operator | CC BY-SA 4.0 |
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| Dec 3, 2023 at 3:10 | history | edited | monotone operator | CC BY-SA 4.0 |
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| Dec 3, 2023 at 2:56 | history | edited | monotone operator | CC BY-SA 4.0 |
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| Dec 3, 2023 at 2:38 | history | asked | monotone operator | CC BY-SA 4.0 |