
Finite time blowup for Lagrangian mods of the three-dimensional Euler equation - cinquemb
https://terrytao.wordpress.com/2016/06/29/finite-time-blowup-for-lagrangian-modifications-of-the-three-dimensional-euler-equation/#comments
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gus_massa
You have linked the comments section. I think it would be better to resubmit
the main page and kill this.

(Note: Killing and resubmitting the same page multiple times make the mods
unhappy, but I think this is a good one time exception to fix one error.)

~~~
cinquemb
I really wanted to link to arXiv:1606.08481v1[0] because its a really fun
paper as he walks through how he sets up the problem (even if he limits it to
rank 3 real spaces, but its fun to think about how it could be represented in
rank arbitrary-n real spaces and ways of computing such), but figured people
might engage more with the discussion on his blog :P

He wrote a couple days ago:

"One possibility is that of a self-similar ansatz, which under suitable
rescaled coordinates would become a steady-state solution of the Euler
equations with an additional scaling correction to the velocity field. Such
self-similar solutions cannot exist if the vorticity is sufficiently localised
(basically because of energy conservation), and there are various results by
Chae relaxing the localisation requirement, however as far as I know a self-
similar solution that is slowly decaying in space has not yet been ruled out.
As I remarked in the paper, I attempted to construct such a self-similar
solution, but was not able to do so (particularly if one wanted to impose
self-adjointness on the vector potential operator, as this together with self-
similarity imposed a lot of constraints on the solution)."

And it made me think about how we can go about computing such, and since
lately I've been kind of obsessed with the Jacobi-Davison algo, and how it
uses the ritz method (to compute the ritz ansatz), could maybe fit here in
meeting the conditions of self-adjointness on the vector potential operator
without imposing constraints on the solution as it exists? In armadillio's
svds[1], they embed their "vector potential operator" matrix X in larger
dimensional sparse space reflected along its diagonal:

⎡ zeros(X.n_rows, X.n_rows) X ⎤

⎣ X.t() zeros(X.n_cols, X.n_cols) ⎦

And this has "worked" for other physical systems I computed things for
(spacial beamforming with eeg), but this specific application and the care for
the mathematical rigor needed to explore more thoughtfully is above my
paygrade :P

[0] [http://arxiv.org/pdf/1606.08481v1](http://arxiv.org/pdf/1606.08481v1)

[1]
[http://arma.sourceforge.net/docs.html#svds](http://arma.sourceforge.net/docs.html#svds)

