Dynamical and thermodynamical stability of two-dimensional flows: variational principles and relaxation equations
Laboratoire de Physique Théorique (CNRS UMR 5152), Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
Corresponding author: a firstname.lastname@example.org
Revised: 26 February 2009
Published online: 9 June 2009
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
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2009