Timeline for Classical theory for the incompressible Euler equation (reference request)
Current License: CC BY-SA 3.0
12 events
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| Apr 23, 2014 at 9:34 | vote | accept | Juhana Siljander | ||
| Apr 15, 2014 at 9:27 | answer | added | Zurab Silagadze | timeline score: 4 | |
| Apr 13, 2014 at 12:58 | comment | added | Juhana Siljander | So in particular, there seems to be a gap in what is known. For instance, if the velocity is in $L^\infty$, but the vorticity only in $L^2$, nothing seems to be known? | |
| Apr 13, 2014 at 12:56 | comment | added | Juhana Siljander | So if I understood correctly, the Beale-Kato-Majda criterion gives regularity under appropriate assumptions on the initial vorticity $\omega_0$ and if $\omega$ is $L^1$ in time and $L^\infty$ in space. But unlike the Serrin result, this is not purely local in the sense that the initial data plays a role in the result. Concerning weak solutions the best result I could find was the one by De Lellis and Székelyhidi that local boundedness does not imply uniqueness. So altogether bounded velocity is not enough, but bounded vorticity is (at least in some sense). | |
| Apr 12, 2014 at 13:55 | comment | added | marcoromito | Something on the line of a ``Serrin result'' is the Beale-Kato-Majda criterion, that holds for Navier-Stokes as well as Euler (not sure if it is on the book, but look for the paper). For local regularity of Euler, well, I do not know. Euler, especially in 3d might be a very bad guy (look at the examples of Scheffer, Shnirelman, and the recent works of De Lellis etc.) | |
| Apr 11, 2014 at 14:22 | comment | added | Juhana Siljander | Actually it seems that the book is available online. I quickly browsed the book and it seems that most of the existence and uniqueness results are global in the sense that one assumes something from the initial velocity/vorticity. So I am wondering are there any purely local regularity results or is this completely hopeless? | |
| Apr 11, 2014 at 14:06 | history | edited | Juhana Siljander | CC BY-SA 3.0 |
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| Apr 11, 2014 at 14:05 | comment | added | Juhana Siljander | Thanks. I have to see whether I can find the book somewhere. I was also able to find some survey papers by googling, but they did not quite answer all of my questions. | |
| Apr 9, 2014 at 12:28 | comment | added | marcoromito | a starting point might be: Majda-Bertozzi: Vorticity and incompressible flow | |
| Apr 8, 2014 at 21:07 | history | edited | Juhana Siljander | CC BY-SA 3.0 |
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| Apr 8, 2014 at 19:58 | history | edited | Juhana Siljander | CC BY-SA 3.0 |
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| Apr 8, 2014 at 18:21 | history | asked | Juhana Siljander | CC BY-SA 3.0 |