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

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 $F_{\mathrm{be}}$ 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 $F_{\mathrm{be}}(\{0\})$ 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.

This is the second renew version of the original paper that has been appeared in arXiv and published in J.Stat.Phys

References in corpus (7)

• Kinetics of Bose-Condensation
• Condensation of bosons in kinetic regime
• On the quantum Boltzmann equation
• Not to normal order - Notes on the kinetic limit for weakly interacting quantum fluids
• On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
• On the Cauchy problem for the homogeneous Boltzmann-Nordheim equation for bosons: local existence, uniqueness and creation of moments
• On low temperature kinetic theory; spin diffusion, Bose Einstein condensates, anyons

Cited by in corpus (3)

• On semi-classical limit of spatially homogeneous quantum Boltzmann equation: weak convergence
• On the spatially homogeneous Boltzmann equation for Bose-Einstein particles with balanced potentials
• Wave turbulence and collective behavior models for wave equations with short- and long-range interactions