In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad’s angular cutoff assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cut-off case and conjecture what we believe to be the right rate of convergence in that case.

On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials

Bertrand Lods
2018-01-01

Abstract

In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad’s angular cutoff assumption. This is done by an adaptation of the famous entropy method and its variants, resulting in explicit algebraic, or even stretched exponential, rates of convergence to equilibrium under appropriate assumptions. The novelty in our approach is that it involves functional inequalities relating the entropy to its production rate, which have independent applications to equations with mixed linear and non-linear terms. We also briefly discuss some properties of the equation in the non-cut-off case and conjecture what we believe to be the right rate of convergence in that case.
2018
462
801
839
https://www.sciencedirect.com/science/article/pii/S0022247X17311307
Functional inequalities; Entropy; Boltzmann equation; Soft potentials
José Canizo; Amit Einav; Bertrand Lods
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1661630
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