On the $L^2$-instability and $L^2$-controllability of steady flows of an ideal incompressible fluid

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): 1-8. <http://eudml.org/doc/93370>.

@article{Shnirelman1999,
abstract = {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.},
author = {Shnirelman, Alexander},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-8},
publisher = {Université de Nantes},
title = {On the $\{L\}^2$-instability and $\{L\}^2$-controllability of steady flows of an ideal incompressible fluid},
url = {http://eudml.org/doc/93370},
year = {1999},
}

TY - JOUR
AU - Shnirelman, Alexander
TI - On the ${L}^2$-instability and ${L}^2$-controllability of steady flows of an ideal incompressible fluid
JO - Journées équations aux dérivées partielles
PY - 1999
PB - Université de Nantes
SP - 1
EP - 8
AB - 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.
LA - eng
UR - http://eudml.org/doc/93370
ER -