Timeline for How to prove the existence of weak solutions of parabolic PDEs using Rothe's method?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 19, 2021 at 13:51 | comment | added | Wentao Hu | @mlk Thank you very much! I'll try it. | |
| May 12, 2021 at 7:16 | comment | added | mlk | Because of the elliptic bounds $\sqrt{\sum_{ij} a_{ij} \partial_i u \partial_j u}$ behaves like an equivalent norm to $\|\nabla u\|$. So if you work with that directly you avoid all those $\lambda$ and $M$ factors that ruin your estimate. The same with $cu^2$. Of course it is made a bit more difficult by the fact that $a$ and $c$ are time-dependent, which is where some additional terms come in. | |
| May 12, 2021 at 4:35 | comment | added | Wentao Hu | @mlk Thank you very much! Could you please explain what "the norm induced by $a_{ij}$" means? Forgive me for being stupid. | |
| May 10, 2021 at 14:36 | comment | added | mlk | I am more familiar with the related variational approach of "minimizing movements", so I don't know how to proceed precisely, but there the key is to work with the correct energy inequality. I.e. in particular you want to do everything until (11), with the norm induced by $a_{ij}$ instead of $\|\nabla u\|^2$. Only after iterating you use the elipticity once. This way the constant will not depend on $m$ (you might get an (exponential) dependence on $hm$, but that is bounded by $T$ and thus fine). | |
| May 10, 2021 at 12:55 | comment | added | Wentao Hu | Also posted in math.stackexchange.com/questions/4133845/… | |
| May 10, 2021 at 12:53 | history | asked | Wentao Hu | CC BY-SA 4.0 |