Existence results for mean field equations with turbulence

TL;DR

This paper proves the existence of solutions for a class of mean field equations modeling two-dimensional turbulence, using variational methods and min-max schemes on Riemannian surfaces.

Contribution

It establishes new existence results for mean field equations with turbulence parameters, extending previous work through a variational approach and min-max techniques.

Findings

Solutions exist when parameters satisfy specific inequalities.

The variational approach effectively handles the nonlinearity.

Results apply to general closed Riemannian surfaces.

Abstract

In this paper we consider the following form of the so-called Mean field equation arising from the statistical mechanics description of two dimensional turbulence \begin{equation}\label{eq:study} - \D_g u = \rho_1 (\frac{e^{u}}{\int_\Sig e^{u} dV_g}-1)-\rho_2 (\frac{e^{-u}}{\int_\Sig e^{-u} dV_g} - 1) \end{equation} on a given closed orientable Riemannian surface ($\Sigma, g$) with volume 1, where $\rho_1, \rho_2$ are real parameters. Exploiting the variational structure of the problem and running a min-max scheme introduced by Djadli and Malchiodi, we prove that if $k$ is a positive integer, $\rho_1$ and $\rho_2$ two real numbers such that $\rho_1\in (8k\pi, 8(k+1)\pi)$ and $\rho_2<4\pi$ then $\eqref{eq:study}$ is solvable.