Blow-up analysis for some mean field equations involving probability measures from statistical hydrodynamics

TL;DR

This paper investigates the blow-up behavior of solutions to a class of mean field equations involving probability measures, relevant to statistical hydrodynamics, and explores related inequalities on Riemannian surfaces.

Contribution

It provides a detailed blow-up analysis for mean field equations with probability measures, extending understanding of their solution behavior and associated inequalities.

Findings

Characterization of blow-up phenomena for solution sequences

Extension of Trudinger-Moser inequality in this context

Identification of conditions for solution existence and blow-up

Abstract

Motivated by the mean field equations with probability measure derived by Sawada-Suzuki and by Neri in the context of the statistical mechanics description of two-dimensional turbulence, we study the semilinear elliptic equation with probability measure: {equation*} -\Delta v=\lambda\int_I V(\alpha,x,v)e^{\alpha v}\,\Pda -\frac{\lambda}{|\Omega|}\iint_{I\times\Om}V(\alpha,x,v)e^{\alpha v}\,\Pda dx, {equation*} defined on a compact Riemannian surface. This equation includes the above mentioned equations of physical interest as special cases. For such an equation we study the blow-up properties of solution sequences. The optimal Trudinger-Moser inequality is also considered.