Timeline for Local smoothness of harmonic heat flow
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jul 18, 2023 at 15:02 | comment | added | Luis Yanka Annalisc | @RBega2 I find that for weak solutions above, I cannot get the monotonicity formula. I have searched for pseudo-locality but the materials are not meet my requirement. Can you give me more precise references? | |
| Jul 12, 2023 at 17:10 | comment | added | RBega2 | I don't know a reference off the top of my head for harmonic map heat flow, but this is basically an example of pseudo-locality (which there are a number of references for in the context of mean curvature flow or Ricci flow). The basic idea is that as the initial data is smooth on sufficiently small scales the energy is not concentrated and so a singularity can't form until a definite amount of time passes. Of course there can be singularities after some amount of time so you only get smoothness for small $T$ | |
| Jul 12, 2023 at 15:33 | history | edited | Daniel Asimov | CC BY-SA 4.0 |
Fixed a few words
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| Jul 12, 2023 at 15:20 | comment | added | Luis Yanka Annalisc | @RBega2 I know that the blow up theory can estimate the Hausdorff measure of singular points but I do not know how to combine this argument and the smoothness of the initial data to eliminate the sigularities. Can you give me some references or hints? | |
| Jul 12, 2023 at 15:18 | history | edited | Luis Yanka Annalisc | CC BY-SA 4.0 |
added 28 characters in body
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| Jul 10, 2023 at 13:17 | comment | added | RBega2 | One possible way is to use the monotonicity formula (due to Struwe) and the smoothness of initial data and smallness of $T$ to show there are no singularities in $(0,T]$ by a blow up analysis. If you want smoothness up to time $0$ an additional argument is required. If the domain is two-dimensional Struwe has an approach based on the energy method which is perhaps a bit simpler. You could also try and establish uniqueness for the weak solution (though this is a delicate issue) and then use the regularity of Lin-Wang -- though maybe you don't like this. | |
| Jul 10, 2023 at 8:20 | history | asked | Luis Yanka Annalisc | CC BY-SA 4.0 |