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Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as ..."
Abstract

Cited by 16 (8 self)
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We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinitedimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the twodimensional stochastic NavierStokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic NavierStokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1
Statistical mechanics of twodimensional and geophysical flows
, 2011
"... The theoretical study of the selforganization of twodimensional and geophysical turbulent flows is addressed based on statistical mechanics methods. This review is a selfcontained presentation of classical and recent works on this subject; from the statistical mechanics basis of the theory up to ..."
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The theoretical study of the selforganization of twodimensional and geophysical turbulent flows is addressed based on statistical mechanics methods. This review is a selfcontained presentation of classical and recent works on this subject; from the statistical mechanics basis of the theory up to applications to Jupiter’s troposphere and ocean vortices and jets. Emphasize has been placed on examples with available analytical treatment in order to favor better understanding of the physics and dynamics. After a brief presentation of the 2D Euler and quasigeostrophic equations, the specificity of twodimensional and geophysical turbulence is emphasized. The equilibrium microcanonical measure is built from the Liouville theorem. Important statistical mechanics concepts (large deviations, mean field approach) and thermodynamic concepts (ensemble inequivalence, negative heat capacity) are briefly explained and described. On this theoretical basis, we predict the output of the long time evolution of complex turbulent flows as statistical equilibria. This is applied to make quantitative models of twodimensional turbulence, the Great Red Spot and other Jovian vortices, ocean jets like the Gulf
and
"... We continue our study on mathematical justification of the emergence of large scale coherent structure in a two dimensional fluid system under small scale random bombardments. We treat the case of small scale random bombardments at discrete times which is different from our earlier work [16] where w ..."
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We continue our study on mathematical justification of the emergence of large scale coherent structure in a two dimensional fluid system under small scale random bombardments. We treat the case of small scale random bombardments at discrete times which is different from our earlier work [16] where we approximated the small scale random kicks by a continuous in time random process. In the absence of geophysical effects, the large scale structure emerging out of the small scale random forcing is the same as the case of continuous in time forcing that we studied before.
unknown title
"... Noname manuscript No. (will be inserted by the editor) Kinetic theory of jet dynamics in the stochastic barotropic and 2D NavierStokes equations ..."
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Noname manuscript No. (will be inserted by the editor) Kinetic theory of jet dynamics in the stochastic barotropic and 2D NavierStokes equations
Statistical mechanics of twodimensional and geophysical flows
"... The theoretical study of the selforganization of twodimensional and geophysical turbulent flows is addressed based on statistical mechanics methods. This review is a selfcontained presentation of classical and recent works on this subject; from the statistical mechanics basis of the theory up to ..."
Abstract
 Add to MetaCart
The theoretical study of the selforganization of twodimensional and geophysical turbulent flows is addressed based on statistical mechanics methods. This review is a selfcontained presentation of classical and recent works on this subject; from the statistical mechanics basis of the theory up to applications to Jupiter’s troposphere and ocean vortices and jets. Emphasize has been placed on examples with available analytical treatment in order to favor better understanding of the physics and dynamics. After a brief presentation of the 2D Euler and quasigeostrophic equations, the specificity of twodimensional and geophysical turbulence is emphasized. The equilibrium microcanonical measure is built from the Liouville theorem. Important statistical mechanics concepts (large deviations, mean field approach) and thermodynamic concepts (ensemble inequivalence, negative heat capacity) are briefly explained and described. On this theoretical basis, we predict the output of the long time evolution of complex turbulent flows as statistical equilibria. This is applied to make quantitative models of twodimensional turbulence, the Great Red Spot and other Jovian vortices, ocean jets like the Gulf