Blow up analysis for Boltzmann-Poisson equation in Onsager's theory for point vortices with multi-intensities

TL;DR

This paper investigates the blow-up behavior of solutions to the Boltzmann-Poisson equation in Onsager's vortex theory, providing estimates and conditions for blow-up in multi-intensity point vortex models.

Contribution

It introduces new blow-up estimates and conditions for the mean field limit of point vortices with multiple intensities, extending existing results to more complex vortex configurations.

Findings

Derived estimates for solution behavior near blow-up points

Established sufficient conditions for blow-up in two-intensity cases

Extended blow-up analysis to standard mean field equations

Abstract

In this paper we consider the minimizing sequence for some energy functional of an elliptic equation associated with the mean field limit of the point vortex distribution one-sided Borel probability measure. If such a sequence blows up, we derive some estimate which is related to the behavior of solution near the blow-up point. Moreover, we study the two intensities case to consider the sufficient condition for this estimate. Our main results are new for the standard mean field equation as well.