Timeline for Does the fast diffusion equation (or singular PME) on a manifold lose mass if the exponent is small enough?
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| Oct 19, 2015 at 13:53 | comment | added | Delio Mugnolo | @AIC: I see. But take a look e.g. at sciencedirect.com/science/article/pii/S0022123605001291 (seemingly only in the range $m>1$) Most results depend on energy estimates (which, given smoothness of the manifolds, will only depend on classical Green formulae or rather on suitably weak definitions of the operators) and suitable Sobolev inequalities. A similar strategy may work in your case, too, and it seems to me that the real challenge is to deduce log-Sobolev inequalities in the manifold case. | |
| Oct 19, 2015 at 13:21 | history | edited | AlC | CC BY-SA 3.0 |
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| Oct 19, 2015 at 13:14 | comment | added | AlC | @DelioMugnolo Singular means the exponent in the nonlineary, which I wrote as $m$ lies in $(0,1)$. The "normal" or "usual" PME has exponent $m \in (1,\infty)$. The PME book focuses only on the $m \in (1,\infty)$ case (see the introduction). | |
| Oct 19, 2015 at 13:09 | comment | added | Delio Mugnolo | @AIC Sorry, I don't get it. What do you call "singular"? The whole book is about the PME! The quoted theorem deals with an even more general setting. (Sorry, I oversaw "closed") | |
| Oct 19, 2015 at 12:20 | comment | added | AlC | @DelioMugnolo It is a closed manifold so no boundary. I did try the book by Vazquez, however, there is no discussionn of the singular case there. | |
| Oct 19, 2015 at 11:05 | comment | added | Delio Mugnolo | Does the manifold have boundary? What about the boundary conditions? No hope of mass conservation if you impose Dirichlet b.c., I suppose. Anyway, a starting point might be §11.5 in J.L. Vázquez' The porous medium equation, Oxford UP 2007 | |
| Oct 19, 2015 at 10:33 | history | asked | AlC | CC BY-SA 3.0 |