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.

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.

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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},
eid = {13},
publisher = {Universit\'e de Nantes},
year = {1999},
mrnumber = {1718998},
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language = {en},
url = {http://www.numdam.org/item/JEDP_1999____A13_0/}
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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/

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