A Harnack-type inequality for a perturbed singular Liouville Equation

TL;DR

This paper establishes a Harnack-type inequality for solutions to a perturbed Liouville equation with singularities, motivated by turbulence models in statistical mechanics, providing bounds on solution behavior near singular points.

Contribution

It introduces a novel Harnack inequality for a class of perturbed Liouville equations with singularities, extending previous results to include perturbations and variable coefficients.

Findings

Proves a Harnack inequality for solutions with singular perturbations.

Shows uniform bounds for solutions near singularities.

Extends classical Liouville equation results to perturbed, singular cases.

Abstract

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we obtain a Harnack-type inequality for sequences of solutions of the following perturbed Liouville equation, \begin{equation}\nonumber -\Delta v_n=\left({\epsilon_n^2+|x|^2}\right)^{\alpha_n}V_n(x)e^{\displaystyle v_n} \qquad\text{in} \,\,\, \Omega, \end{equation} where $\epsilon_n\to0^+$, $\alpha_n\to\alpha_\infty\in(-1,1)$, $\Omega$ is a bounded domain in $\mathbb{R}^2$ containing the origin and $V_n$ satisfies, \begin{equation}\nonumber 0<a\leq V_n\leq b<+\infty, \,\, V_n\in C^{0}(\Omega), \,\,V_n\to V \,\, \text{locally uniformly in}\,\,{\Omega}. \end{equation}