Condensation phenomena in nonlinear drift equations

TL;DR

This paper investigates measure-valued solutions to nonlinear drift equations modeling Bose-Einstein particle concentration, proving finite-time blow-up and mass concentration at the origin, with solutions converging to total mass over time.

Contribution

It establishes existence, uniqueness, and blow-up behavior of solutions to nonlinear drift equations in one dimension, using novel scaling and pseudo-inverse methods.

Findings

Solutions blow up in finite time

Mass concentrates at the origin

Solutions converge to total mass over time

Abstract

We study nonnegative, measure-valued solutions to nonlinear drift type equations modelling concentration phenomena related to Bose-Einstein particles. In one spatial dimension, we prove existence and uniqueness for measure solutions. Moreover, we prove that all solutions blow up in finite time leading to a concentration of mass only at the origin, and the concentrated mass absorbs increasingly the mass converging to the total mass as time goes to infinity. Our analysis makes a substantial use of independent variable scalings and pseudo-inverse functions techniques.