Timeline for PDE: compactness vs blowup
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 13 at 13:05 | answer | added | Lorenzo Pompili | timeline score: 1 | |
| Feb 13 at 12:17 | comment | added | Sebastian Bechtel | @LorenzoPompili I'm aware that the question is not really sharp. Your comment highlights an advantage of the compactness approach: it can yield existence in situations in which there is not uniqueness (you don't get a full global well-posedness result then, of course). | |
| Feb 12 at 14:41 | comment | added | Lorenzo Pompili | Also, compactness methods can also work on the Eucledian space, not just on bounded domains. The main difference between the two approaches is that compactness methods alone do not give you the uniqueness of the solution. So, even if you can prove that solutions exist for all times, the solution could still "blow up" at some time, meaning that it ceases to be unique. And this can also happen when you can prove l.w.p.: for example, nobody knows whether Leray solutions of 3D Navier-Stokes (global weak solutions found via compactness) are unique or not, even though local well-posedness is known. | |
| Feb 12 at 14:33 | comment | added | Lorenzo Pompili | The two things are not disjoint. Proving local well-posedness often involves some compactness method. This usually happens for quasilinear equations, e.g. Euler's equation. | |
| Feb 12 at 14:19 | history | asked | Sebastian Bechtel | CC BY-SA 4.0 |