Blow-up rate estimates for the solutions of the bosonic Boltzmann-Nordheim equation

TL;DR

This paper investigates the blow-up behavior of solutions to the bosonic Boltzmann-Nordheim equation, providing estimates on blow-up rates, conditions for boundedness, and establishing local existence for measure-valued solutions.

Contribution

It offers new blow-up rate estimates, bounds solutions near critical singularities, and proves local existence for unbounded densities in the bosonic Boltzmann-Nordheim equation.

Findings

Solutions blow up at rates estimated near critical singularity

Bounded solutions remain bounded in the uniform norm under certain conditions

Local existence established for measure-valued solutions with unbounded densities

Abstract

In this paper we study the behavior of a class of mild solutions of the homogeneous and isotropic bosonic Boltzmann-Nordheim equation near the blow-up. We obtain some estimates on the blow-up rate of the solutions and prove that, as long as a solution is bounded above by the critical singularity 1/x (the equilibrium solutions behave like this power law near the origin), it remains bounded in the uniform norm. In the last section of the paper, we also prove a local existence result for a class of measure-valued mild solutions, which allows us to solve the Boltzmann-Nordheim equation for some classes of unbounded densities.