The Boltzmann Equation for Bose-Einstein Particles: Condensation in Finite Time

The paper considers the problem of the Bose-Einstein condensation in finite time for isotropic distributional solutions of the spatially homogeneous Boltzmann equation for Bose-Einstein particles with the hard sphere model. We prove that if the initial datum of a solution is a function which is singular enough near the origin (the zero-point of particle energy) but still Lebesgue integrable (so that there is no condensation at the initial time), then the condensation continuously starts to occur from the initial time to every later time. The proof is based on a convex positivity of the cubic collision integral and some properties of a certain Lebesgue derivatives of distributional solutions at the origin. As applications we also study a special type of solutions and present a relation between the conservation of mass and the condensation.