The European Physical Journal B

, Volume 70, Issue 1, pp 73–105 | Cite as

Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations

Topical issue on Generalized Entropies and Non-Linear Kinetics

Abstract

We review and connect different variational principles that have been proposed to settle the dynamical and thermodynamical stability of two-dimensional incompressible and inviscid flows governed by the 2D Euler equation. These variational principles involve functionals of a very wide class that go beyond the usual Boltzmann functional. We provide relaxation equations that can be used as numerical algorithms to solve these optimization problems. These relaxation equations have the form of nonlinear mean field Fokker-Planck equations associated with generalized “entropic” functionals [P.H. Chavanis, Eur. Phys. J. B 62, 179 (2008)].

PACS

05.20.-y Classical statistical mechanics 05.45.-a Nonlinear dynamics and chaos 05.90.+m Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems 47.10.-g General theory in fluid dynamics 47.20.-k Flow instabilities 47.32.-y Vortex dynamics; rotating fluids 

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Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique (CNRS UMR 5152), Université Paul SabatierToulouseFrance

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