In the existing stability theory of steady flows of an ideal incompressible fluid, formulated by V. Arnold, the stability is understood as a stability with respect to perturbations with small in ${L}^{2}$ vorticity. Nothing has been known about the stability under perturbation with small energy, without any restrictions on vorticity; it was clear that existing methods do not work for this (the most physically reasonable) class of perturbations. We prove that in fact, every nontrivial steady flow is unstable in ${L}^{2}$; moreover, every flow may be transformed into any other one, with the same energy and momentum, with the help of an appropriately chosen perturbation with arbitrary small energy. This phenomenon reminds the Arnold’s diffusion. This result is proven by the direct construction of a growing perturbation, which is done by a variational method.

@article{JEDP_1999____A13_0, author = {Shnirelman, Alexander}, title = {On the ${L}^2$-instability and ${L}^2$-controllability of steady flows of an ideal incompressible fluid}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, publisher = {Universit\'e de Nantes}, year = {1999}, mrnumber = {1718998}, zbl = {01810586}, mrnumber = {2000h:76082}, language = {en}, url = {http://www.numdam.org/item/JEDP_1999____A13_0} }

Shnirelman, Alexander. On the ${L}^2$-instability and ${L}^2$-controllability of steady flows of an ideal incompressible fluid. Journées équations aux dérivées partielles (1999), article no. 13, 8 p. http://www.numdam.org/item/JEDP_1999____A13_0/

[A1] Sur la Géométrie diffé rentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1966), 316-361. | Numdam | MR 34 #1956 | Zbl 0148.45301

,[A2] On the a priori estimate in the theory of hydrodynamical stability, Amer. Math. Soc. Transl. 19 (1969), 267-269. | Zbl 0191.56303

,[A3] Mathematical methods of classical mechanics, Springer-Verlag, New York, 1989. | MR 90c:58046 | Zbl 0386.70001

,[A-K] Topological methods in hydrodynamics, Applied Mathematical Sciences, v. 125, Springer-verlag, 1998. | MR 99b:58002 | Zbl 0902.76001

, ,[B] The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Math. Soc. 2 (1989), no. 2, 225-255. | MR 90g:58012 | Zbl 0697.76030

,[M-P] Mathematical theory of incompressible nonviscous fluids, Applied Mathematical Sciences, v. 96, Springer-Verlag, 1994. | MR 94k:76001 | Zbl 0789.76002

, ,[S] The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Math. USSR Sbornik 56 (1987), no. 1, 79-105. | Zbl 0725.58005

,