On the Trend to Equilibrium for Some Dissipative Systems with Slowly Increasing a Priori Bounds

By Giuseppe Toscani, Cédric Villani. 2000

111 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/767951a9fd68b621f9c08947d43b96b7d9086ff3

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates (2003, background): “Nearly all of the forthcoming discussion follows Otto and Villani [28].”
Hypocoercive Diffusion Operators (2006, background): “Among a large literature, I shall only quote some of my own works: entropy production estimates for the spatially homogeneous Boltzmann equation, in collaboration with Giuseppe Toscani [25], [26], [28]; and for certain nonlinear diffusion equations with a convex mean-field interaction, in collaborat”
Exponential decay toward equilibrium via entropy methods for reaction–diffusion equations (2006, background): “), integral equations (such as the spatially homogeneous Boltzmann equation [24, 25, 26]), or kinetic equations ([4], [10, 11], [13]).”
Cercignani's Conjecture is Sometimes True and Always Almost True (2003, background): “The interest to have a nice control of the dependence on Ce upon the various bounds above is demonstrated in [ 40 ].”
On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation (2005, background): “All this analysis can be recast for the Landau collision operator (8), only in simpler terms, since positivity estimates are not even necessary [20,53].”
On the trend to global equilibrium in spatially inhomogeneous entropy‐dissipating systems: The linear Fokker‐Planck equation (2001, background): “Algebraic decay with a fixed, explicit rate is also proven for a very broad class of cross-sections, see [16, 35, 36].”
Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory (2002, methodology): “We note that this theorem does not cover the interesting case β = 3 (Coulomb potential in dimension 3) : the proof in [429] just fails for this limit exponent.”
Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds (2008, background): “[34]), as well as interpolation between an explicit decay in weak norm and controlled growth in strong norm (cf.”
Factorization for non-symmetric operators and exponential H-theorem (2010, result): “.6. A similar result has been proved in [90, Theorem 3] under the additional and fundamental assumptions that f in is non negative and has nite energy and entropy. Moreover the decay rate obtained in [90] was only of order t (k 2)=(2(2 )) and remains valid for 2(0;1). 3.2. Proof of the main results.”
Entropy Dissipation Methods for Degenerate ParabolicProblems and Generalized Sobolev Inequalities (2001, background): “The case of a subquadratic growth of Vx (like jxj ; < 2) has been recently investigated in [63] for the linear case f u u.”
ON THE TREND TO EQUILIBRIUM FOR THE FOKKER-PLANCK EQUATION : AN INTERPLAY BETWEEN PHYSICS AND FUNCTIONAL ANALYSIS (2004, background): “This is not the case : a simple way to overcome the absence of logarithmic Sobolev inequality is exposed in [37].”