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 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.
Highlights
This work is concerned with the asymptotic behaviour of the linear homogeneous Boltzmann equation in the less explored case of soft potential interactions, and with a cut-off assumption
We are interested in the application of entropy techniques to study the approach to equilibrium in the relative entropy sense, and in the application of entropy inequalities to estimate its rate
Our motivation comes partly from the study of the linear Boltzmann equation itself, which is a basic model in kinetic theory describing the collisional interaction of a set of particles with a thermal bath at a fixed temperature
Summary
This work is concerned with the asymptotic behaviour of the linear homogeneous Boltzmann equation in the less explored case of soft potential interactions, and with a cut-off assumption (the precise definition of all the above will be given shortly). Where C(f ) is an explicit functional involving norms of f in appropriate L1κ1 and L1κ2 log L spaces, for a suitable κ1, κ2 To use this inequality to deduce Theorem 1.4 one needs to control C(f ) along the flow of the equation. This is done by an adaptation of similar results from [33]. The last pages of the paper are dedicated to several Appendices that provide additional details that we felt would hinder the flow of the main work
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
