Timeline for Boundary energy estimate of wave equations
Current License: CC BY-SA 3.0
12 events
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| Feb 19, 2015 at 8:33 | comment | added | Willie Wong | If you just want to gain regularity: commute only with rotations and time-translations. The PDE is hyperbolic and you can solve for radial derivatives in terms of the other two. In my comment above the radial field is used as a multiplier, not a commutator, so you don't have to worry. | |
| Feb 19, 2015 at 0:29 | comment | added | Yun K | @WillieWong This is a problem arises from my current work on the free boundary problem of a fluid. What I'm really looking for is the connection between interior and boundary estimate. The inequality provided by Dr. Serre is exactly the type of inequality I'm looking for.In addition, I have tried to commute with rotational vector fields and apply classical energy estimate, since rotational fields preserves the boundary condition. However, I'm a bit worrying about the radial vector field, since it is in the normal direction of the boundary. How can I deal with that? | |
| Feb 19, 2015 at 0:05 | comment | added | Yun K | @WillieWong Thanks for your comments. You are definitely right, one cannot apply K-S on compact domains. What I really meant to say "if we are working in the $\mathbb{R}^{n}$, the we can conclude lifespan estimate via K-S." | |
| Feb 19, 2015 at 0:00 | vote | accept | Yun K | ||
| Feb 19, 2015 at 0:00 | history | edited | Yun K | CC BY-SA 3.0 |
deleted 66 characters in body
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| Feb 18, 2015 at 10:06 | comment | added | Willie Wong | On the other hand, you can get very crude estimates on $\| \partial_r u\|_{L^2([0,T]\times \partial D)}$ by using the multiplier field $r \partial_r = \sum x_i \partial_i$ and the fundamental energy estimate. To get higher regularity commute the equation by the angular momentum vector fields. It would be better if you can say something about what types of estimates are you looking for, or possibly what applications you are interested in. | |
| Feb 18, 2015 at 9:55 | comment | added | Willie Wong | Also, by what do you mean the boundary energy? By your assumptions only the normal derivative is non-vanishing on $\partial D$ so it seems a bit strange to write that as $\|u (t,\cdot)\|_{H^r(\partial D)}$, since $u|_{\mathbb{R}\times\partial D} \equiv 0$. | |
| Feb 18, 2015 at 9:42 | comment | added | Willie Wong | Life span with Klainerman-Sobolev? What the hey? On a compact domain you have no dispersive decay and Klainerman-Sobolev absolutely cannot be applied. Besides, you are working with a linear inhomogeneous equation so as long as $F$ is sufficiently regular your classical solution exists for all time. On a compact domain you don't really want to commute with vector fields, since if the vector fields are transverse to the boundary it interchanges Dirichlet and Neumman boundary conditions. You would be better off commuting the Dirichlet Laplacian instead to get spatial regularity. | |
| Feb 18, 2015 at 9:27 | answer | added | Denis Serre | timeline score: 3 | |
| Feb 18, 2015 at 5:32 | history | edited | Joonas Ilmavirta | CC BY-SA 3.0 |
edited body; edited tags
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| Feb 18, 2015 at 5:08 | review | First posts | |||
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| Feb 18, 2015 at 5:07 | history | asked | Yun K | CC BY-SA 3.0 |