Timeline for Classical solutions for parabolic PDE's
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8 events
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| Nov 26, 2017 at 23:30 | comment | added | user21574 | Existence of such solutions correspond to stability of relative tangent sheaf $(T_{X/\Delta})^{**}$ in the sense of Mumford. See the last page slideshare.net/HassanJolany/canonical-metric-on-khler-manifolds | |
| Nov 26, 2017 at 23:27 | comment | added | user21574 | But in such variant case, we need to rune the flow by two times $s$ and $t$ such that the $s\to 0$ while $t\to\infty$ and $s$ comes from the fibers $\pi^{-1}(s):=X_s$. I think the right flow is hyperbolic type PDE as I wrote in my previous comment. Such flow is the generalized ersion of my question here which I posted 5 years ago mathoverflow.net/questions/106142/… | |
| Nov 26, 2017 at 23:21 | comment | added | user21574 | If you want to consider the variation of Kahler Ricci flow on degeneration of Kahler-Einstein metrics $\pi:X\to \Delta$, then we need to resolve singularities by two times $s$ and $t$, i.e, $$\frac{\partial^2 \omega(s,t)}{\partial s\partial t}=-Ric_{X/\Delta}(s,t)-f(s)\omega(s,t)$$, which is hyperbolic PDE in this case we need to the positivity of initial metric and I am not sure the statement of Deane Yang is true in this case. where $f(s)$ is fiberwise constant. In KRF case we need to run the flow by one time $t$, | |
| Nov 19, 2017 at 1:15 | comment | added | Deane Yang | I've been looking for this myself. | |
| Nov 18, 2017 at 17:30 | comment | added | L.F. Cavenaghi | thanks for answering @DeaneYang, could you give me some reference? | |
| Nov 18, 2017 at 0:47 | comment | added | Deane Yang | If the initial data is $C^2$, then the solution is a smooth and therefore classic solution for $0 \le t < T$ for some $T > 0$. If the initial data is less smooth, then it will be a smooth classic solution at least for $0 < t < T$. | |
| Nov 17, 2017 at 22:49 | history | edited | L.F. Cavenaghi |
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| Nov 17, 2017 at 21:49 | history | asked | L.F. Cavenaghi | CC BY-SA 3.0 |