On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials
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Functional inequalitiesEntropyBoltzmann equationSoft potentials
Cañizo Rincón, J.A.; Einav, A.; Lods, B. On the rate of convergence to equilibrium for the linear Boltzmann equation with soft potentials. J. Math.Anal.Appl.462(2018)801–839. [https://doi.org/10.1016/j.jmaa.2017.12.052].
SponsorshipJAC was supported by project MTM2014-52056-P, funded by the Spanish government and the European Regional Development Fund. AE was partially supported by EPSRC grant EP/L002302/1 and partially supported by the Austrian Science Fund (FWF) grant M 2104-N32.
In this work we present several quantitative results of convergence to equilibrium for the linear Boltzmann operator with soft potentials under Grad’s angular cut-off 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.