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On the \(L^2\)-instability and \(L^2\)-controllability of steady flows of an ideal incompressible fluid. (English) Zbl 1103.76328
Proceedings of the conference on partial differential equations, Saint-Jean-de-Monts, France, May 31–June 4, 1999. Exp. Nos. I–XIX (1999). Nantes: Université de Nantes (ISBN 2-86939-146-3/pbk). Exp. No. XIII, 8 p. (1999).
Summary: n the existing stability theory of steady flows of an ideal incompressible fluid, formulated by V. Arnold, stability is understood as stability with respect to perturbations with vorticity small in \(L^2\). Nothing has been known about the stability under perturbations 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 one of the Arnold diffusion. This result is proven by the direct construction of a growing perturbation, which is done by a variational method.
For the entire collection see [Zbl 0990.00047].
76E99 Hydrodynamic stability
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76B75 Flow control and optimization for incompressible inviscid fluids
93B05 Controllability
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