5 ms·

Finite time blowup for Lagrangian mods of the three-dimensional Euler equation

1 points by cinquemb 10y ago

gus_massa 10y ago
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 10y ago
  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

1 points by cinquemb 10y ago

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 10y ago
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