The main issue of this note is the study of the long time behavior of 2d perfect fluids on the 2-torus governed by the incompressible Euler equations and the genericity of small scale creation in said limit. Its aim is twofold: first introduce non specialists to some key conjectures in the field today due to Shnirelman, Šverák and Yudovich respectively as well as some results towards those conjectures. Second we present a recent result by the author [19] on the generic character of small creation for the Lagrangian flow which is build upon a pioneering Lyapunov construction due to Shnirelman [20].
@article{SLSEDP_2024-2025____A1_0,
author = {Ayman Rimah Said},
title = {Small scale creation in the long time behavior of 2d~perfect~fluids},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:4},
pages = {1--8},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2024-2025},
doi = {10.5802/slsedp.174},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.174/}
}
TY - JOUR AU - Ayman Rimah Said TI - Small scale creation in the long time behavior of 2d perfect fluids JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:4 PY - 2024-2025 SP - 1 EP - 8 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.174/ DO - 10.5802/slsedp.174 LA - en ID - SLSEDP_2024-2025____A1_0 ER -
%0 Journal Article %A Ayman Rimah Said %T Small scale creation in the long time behavior of 2d perfect fluids %J Séminaire Laurent Schwartz — EDP et applications %Z talk:4 %D 2024-2025 %P 1-8 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.174/ %R 10.5802/slsedp.174 %G en %F SLSEDP_2024-2025____A1_0
Ayman Rimah Said. Small scale creation in the long time behavior of 2d perfect fluids. Séminaire Laurent Schwartz — EDP et applications (2024-2025), Exposé no. 4, 8 p. doi : 10.5802/slsedp.174. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.174/
[1] Jacob Bedrossian and Nader Masmoudi. Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations. Publications mathématiques de l’IHÉS, 122(1):195–300, 2015. | DOI | Zbl
[2] Ronald R Coifman and Yves Meyer. Au delà des opérateurs pseudo-différentiels, volume 57 of Astérisque. Société mathématique de France, 1978. | Zbl
[3] Peter Constantin. On the Euler equations of incompressible fluids. Bulletin of the American Mathematical Society, 44(4):603–621, 2007. | DOI | Zbl
[4] Sergey A Denisov. Infinite superlinear growth of the gradient for the two-dimensional Euler equation, 2009. | arXiv
[5] Theodore D Drivas and Tarek M Elgindi. Singularity formation in the incompressible Euler equation in finite and infinite time, 2022. | arXiv
[6] Theodore D Drivas, Tarek M Elgindi, and In-Jee Jeong. Twisting in Hamiltonian flows and perfect fluids, 2023. | arXiv
[7] Tarek Elgindi, Ryan Murray, and Ayman Said. On the long-time behavior of scale-invariant solutions to the 2d Euler equation and applications, 2022. | arXiv
[8] Patrick Gérard. Mesures semi-classiques et ondes de Bloch. In Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi" Séminaire Goulaouic-Schwartz", pages 1–19. Éditions de l’École polytechnique, 1991. | Zbl
[9] Alexandru Ionescu and Hao Jia. Axi-symmetrization near point vortex solutions for the 2D Euler equation. Communications on Pure and Applied Mathematics, 75(4):818–891, 2022. | DOI | Zbl
[10] Alexandru D Ionescu and Hao Jia. Nonlinear inviscid damping near monotonic shear flows. Acta Math, 230:321–399, 2023. | DOI | MR | Zbl
[11] In-Jee Jeong and Ayman R Said. Logarithmic spirals in 2d perfect fluids, 2023. | arXiv
[12] Boris Khesin, Gerard Misiołek, and Alexander Shnirelman. Geometric hydrodynamics in open problems. Archive for Rational Mechanics and Analysis, 247(2):15, 2023. | DOI | Zbl
[13] Alexander Kiselev and Vladimir Šverák. Small scale creation for solutions of the incompressible two-dimensional Euler equation. Annals of mathematics, 180(3):1205–1220, 2014. | DOI | Zbl
[14] Herbert Koch. Transport and instability for perfect fluids. Mathematische Annalen, 323(3):491–523, 2002. | DOI | Zbl
[15] Pierre-Louis Lions and Thierry Paul. Sur les mesures de Wigner. Revista matemática iberoamericana, 9(3):553–618, 1993. | DOI | Zbl
[16] Nader Masmoudi and Weiren Zhao. Nonlinear inviscid damping for a class of monotone shear flows in finite channel, 2020. | arXiv
[17] Andrey Morgulis, Alexander Shnirelman, and Victor Yudovich. Loss of smoothness and inherent instability of 2D inviscid fluid flows. Communications in Partial Differential Equations, 33(6):943–968, 2008. | DOI | Zbl
[18] Nikolai Semenovich Nadirashvili. Wandering solutions of Euler’s D-2 equation. Functional Analysis and Its Applications, 25(3):220–221, 1991. | DOI | Zbl
[19] Ayman Rimah Said. Small scale creation of the Lagrangian flow in 2d perfect fluids, 2024. | arXiv
[20] Alexander Shnirelman. Evolution of singularities, generalized Liapunov function and generalized integral for an ideal incompressible fluid. American Journal of Mathematics, 119(3):579–608, 1997. | DOI | Zbl
[21] Alexander Shnirelman. Microglobal analysis of the Euler equations. Journal of mathematical fluid mechanics, 7:S387–S396, 2005. | DOI | Zbl
[22] Alexander Shnirelman. On the long time behavior of fluid flows. Procedia IUTAM, 7:151–160, 2013. | DOI
[23] Vladimir Šverák. Course notes. https://www-users.cse.umn.edu/~sverak/course-notes2011.pdf, 2011/2012.
[24] V. Yudovich. The loss of smoothness of the solutions of Euler equations with time. Dinamika Splosn. Sredy Vyp, 16:71–78, 1974.
[25] V. Yudovich. On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid. Chaos: An Interdisciplinary Journal of Nonlinear Science, 10(3):705–719, 2000. | DOI
[26] Victor I Yudovich. Non-stationary flow of an ideal incompressible liquid. USSR Computational Mathematics and Mathematical Physics, 3(6):1407–1456, 1963. | DOI | Zbl
[27] Andrej Zlatoš. Exponential growth of the vorticity gradient for the Euler equation on the torus. Advances in Mathematics, 268:396–403, 2015. | DOI | Zbl
Cité par Sources :