Prolongationally stable discrete flows

Ronald Knight

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In this survey article we discuss some recent results concerning strong spectral estimates for Ruelle transfer operators for contact flows on basic sets similar to these of Dolgopyat obtained in the case of Anosov flows with C1 stable and unstable foliations. Some applications of Dolgopyat's results and the more recent ones are also described.

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

Alexander Shnirelman (1999)

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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...