The Spatially Homogeneous Boltzmann Equation for Bose-Einstein Particles: Rate of Strong Convergence to Equilibrium

TL;DR

This paper proves the strong convergence of solutions to the Bose-Einstein Boltzmann equation towards equilibrium, including the condensation point, with explicit algebraic convergence rates, extending previous weak convergence results.

Contribution

It establishes the strong convergence to equilibrium for all radially symmetric, non-singular initial data, including the condensation point, with explicit algebraic convergence rates.

Findings

Long time convergence of $F_t( ext{0})$ to Bose-Einstein condensation established.

Strong convergence to equilibrium proven for all radially symmetric, non-singular initial data.

Explicit algebraic rate of convergence obtained for arbitrary temperature.

Abstract

The paper is a continuation of our previous work on the spatially homogeneous Boltzmann equation for Bose-Einstein particles with quantum collision kernel that includes the hard sphere model. Solutions $F_t$ under consideration that conserve the mass, momentum, and energy and converge at least weakly to equilibrium $F_{{\rm be}}$ as $t\to\infty$ have been proven to exist at least for radially symmetric and non-singular initial data, and for the case of low temperature, $F_t$ have to be positive Borel measures. The new progress is as follows: we prove that the long time convergence of $F_t(\{0\})$ to the Bose-Einstein condensation $F_{{\rm be}}(\{0\})$ for low temperature holds for all radially symmetric and non-singular initial data $F_0$. This immediately implies the long time strong convergence to equilibrium. We also obtain an algebraic rate of the strong convergence for arbitrary temperature. Our proofs are based on the entropy control, Villani's inequality for the entropy dissipation, a suitable time-dependent convex combination between the solution and a fixed positive function (in order to overcome the lack of positive lower bound), the convex-positivity of the cubic collision integral, and an iteration technique for obtaining a positive lower bound of condensation.