Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states

1990 ◽  
Vol 70 (1) ◽  
pp. 1-47 ◽  
Author(s):  
Yuri Kifer
Keyword(s):  
Stochastic Stability ◽  
Equilibrium States ◽  
Topological Pressure ◽  
Principal Eigenvalues ◽  
Stability Of Equilibrium

ASYMPTOTIC STABILITY OF EQUILIBRIUM STATES FOR AMBIPOLAR PLASMAS

2004 ◽  
Vol 14 (09) ◽  
pp. 1361-1399 ◽  
Author(s):  
V. GIOVANGIGLI ◽  
B. GRAILLE
Keyword(s):  
Partial Differential Equations ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Null Space ◽  
Current Model ◽  
Debye Length ◽  
Equilibrium States ◽  
Electron Mass ◽  
Partial Differential ◽  
Stability Of Equilibrium

We investigate a system of partial differential equations modeling ambipolar plasmas. The ambipolar — or zero current — model is obtained from general plasmas equations in the limit of vanishing Debye length. In this model, the electric field is expressed as a linear combination of macroscopic variable gradients. We establish that the governing equations can be written as a symmetric form by using entropic variables. The corresponding dissipation matrices satisfy the null space invariant property and the system of partial differential equations can be written as a normal form, i.e. in the form of a symmetric hyperbolic–parabolic composite system. By properly modifying the chemistry source terms and/or the diffusion matrices, asymptotic stability of equilibrium states is established and decay estimates are obtained. We also establish the continuous dependence of global solutions with respect to vanishing electron mass.

Tail pressure and the tail entropy function

2011 ◽  
Vol 32 (4) ◽  
pp. 1400-1417 ◽  
Author(s):  
YUAN LI ◽  
ERCAI CHEN ◽  
WEN-CHIAO CHENG
Keyword(s):  
Variational Principle ◽  
Invariant Measures ◽  
Direct Proof ◽  
Equilibrium States ◽  
Entropy Function ◽  
Topological Pressure

AbstractBurguet [A direct proof of the tail variational principle and its extension to maps. Ergod. Th. & Dynam. Sys.29 (2009), 357–369] presented a direct proof of the variational principle of tail entropy and extended Downarowicz’s results to a non-invertible case. This paper defines and discusses tail pressure, which is an extension of tail entropy for continuous transformations. This study reveals analogs of many known results of topological pressure. Specifically, a variational principle is provided and some applications of tail pressure, such as the investigation of invariant measures and equilibrium states, are also obtained.

scholarly journals Equilibrium stability for non-uniformly hyperbolic systems

2018 ◽  
Vol 39 (10) ◽  
pp. 2619-2642 ◽  
Author(s):  
JOSÉ F. ALVES ◽  
VANESSA RAMOS ◽  
JAQUELINE SIQUEIRA
Keyword(s):  
Hyperbolic Systems ◽  
Equilibrium States ◽  
Topological Pressure ◽  
Equilibrium Stability ◽  
Hölder Continuous ◽  
Skew Products ◽  
Uniformly Hyperbolic Systems ◽  
Wide Family

We prove that for a wide family of non-uniformly hyperbolic maps and hyperbolic potentials we have equilibrium stability, i.e. the equilibrium states depend continuously on the dynamics and the potential. For this we deduce that the topological pressure is continuous as a function of the dynamics and the potential. We also prove the existence of finitely many ergodic equilibrium states for non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials. Finally, we show that these equilibrium states vary continuously in the $\text{weak}^{\ast }$ topology within such systems.

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