Timeline for Non-uniqueness of flow for divergence free vector fields
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| S May 24, 2018 at 20:56 | history | bounty ended | CommunityBot | ||
| S May 24, 2018 at 20:56 | history | notice removed | CommunityBot | ||
| May 22, 2018 at 18:12 | answer | added | StheW | timeline score: 0 | |
| May 22, 2018 at 17:06 | comment | added | user111164 | @Bazin Mmm... what do you mean by "good"? I have read Depauw's example, I agree that the construction is rather explicit. The point is that I do not see how to "localize" (this is probably related also to what Uncle Sam was saying). Thanks for the comment. | |
| May 22, 2018 at 16:23 | comment | added | Bazin | @user111164 Well, I think that this requires some work: taking a close look at Depauw's example, you will see that his construction of a vector field $V$ so that $curl V$ does not have uniqueness is rather explicit (as well as Aizenman's). Then you have to check $curl(\phi V)$ where $\phi$ is a cutoff function and carefully follow what happens in the transition region where $0<\phi <1$. Some "good" choices of $\phi$ may simplify matters. | |
| May 22, 2018 at 13:42 | comment | added | Romeo | +1, nice question! I am interested into the Eulerian side and I know Depauw's example. Could one do something similar to what @Bazin suggests also in that case? | |
| May 21, 2018 at 18:57 | comment | added | user111164 | @Bazin Thanks, this is indeed what I was looking for but I am not totally sure of how to do this. I agree that Aizenman's field is a curl and multiply times a test function seems a reasonable thing to do to have (3) satisfied. But how can we be sure that we have non-unique measure preserving flow? Do you see how to fix these details? I would be grateful if you could kindly expand your comment. Thanks. | |
| May 21, 2018 at 9:55 | comment | added | Bazin | It seems that Aizenman's article deals with a non-uniqueness result and not with the existence of two different flows. The Depauw paper provides an Eulerian perspective, i.e. replaces the ODE by a first-order PDE. On the other hand, to get 3 in your question, it is quite likely that Aizenman's vector field $X$ is three-dimensional and thus $X=\text{curl} V$. To get 3, it should be enough to take $\tilde X=\text{curl}(\phi V)$ where $\phi$ is a cutoff function. | |
| May 18, 2018 at 8:54 | answer | added | RaphaelB4 | timeline score: 0 | |
| S May 16, 2018 at 19:44 | history | bounty started | user111164 | ||
| S May 16, 2018 at 19:44 | history | notice added | user111164 | Draw attention | |
| May 16, 2018 at 16:37 | answer | added | Michael Renardy | timeline score: 3 | |
| May 15, 2018 at 9:18 | history | edited | user111164 | CC BY-SA 4.0 |
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| May 14, 2018 at 19:12 | history | edited | user111164 | CC BY-SA 4.0 |
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| May 14, 2018 at 11:58 | history | asked | user111164 | CC BY-SA 4.0 |