Timeline for Global diffeomorphisms of $\mathbb R^n$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 20, 2019 at 13:40 | comment | added | Abdelmalek Abdesselam | Maybe Euler does not have the kind of symmetries you are hoping for, but it has a purely group theoretical formulation due to Arnold, as reviewed in projecteuclid.org/euclid.bams/1183530815 so in a way symmetries are present. | |
| Jun 19, 2019 at 20:10 | comment | added | Bazin | @ Abdelmalek Well, is the invariance group of Euler equation that large? In fact, as far as invariance is concerned, for Euler as well as for NS, you need to give a geometrical meaning to the term $(v\cdot \nabla) v$ by using the Lie derivative along $v$ and also some identification of the cotangent bundle with the tangent bundle, conveniently provided by a Riemannian structure. Of course there are other structures which could provide the identification ; anyhow Euler as well as NS are equation on one-forms. | |
| Jun 19, 2019 at 15:33 | comment | added | Abdelmalek Abdesselam | I see. May I interest you in Euler's equation instead...:) | |
| Jun 19, 2019 at 15:20 | comment | added | Bazin | @ Abdelmalek Abdesslam If $v(x,t)$ is a (vector-field) solution of the incompressible Navier-Stokes equation in $\mathbb R^3$, $\mathbb R\ni\lambda\not=0$ and $\kappa$ is as in my question then $\lambda \kappa'(x)^{-1}v(\lambda \kappa (x), \lambda ^2 t)$ should also be a solution. My motivation was to check that some non-linear transformations could have been missed. It is of course a bit disappointing that the invariance of NSI group looks so small. By the way, I would be interested to know if (and where) a thorough study of that invariance is performed. | |
| Jun 19, 2019 at 15:11 | answer | added | Abdelmalek Abdesselam | timeline score: 2 | |
| Jun 19, 2019 at 15:08 | comment | added | Abdelmalek Abdesselam | In view of the very simple (negative as far as existence of nonlinear examples) answers, this does not look like the proper formulation of the question has been reached. What are you after, intuitively? | |
| Jun 19, 2019 at 13:25 | answer | added | Pietro Majer | timeline score: 8 | |
| Jun 19, 2019 at 13:01 | answer | added | Piotr Hajlasz | timeline score: 6 | |
| Jun 19, 2019 at 11:50 | history | asked | Bazin | CC BY-SA 4.0 |