Supercritical Mean Field Equations on convex domains and the Onsager's statistical description of two-dimensional turbulence

TL;DR

This paper investigates the existence, multiplicity, and structure of solutions to supercritical mean field equations on convex domains, advancing the understanding of two-dimensional turbulence and the Onsager's statistical model.

Contribution

It provides the first existence results for supercritical mean field equations on convex domains and analyzes solution structure and uniqueness in thin domains.

Findings

Existence of solutions in supercritical regime on convex, thin domains.

Uniqueness of solutions with bounded energy on thin domains.

Asymptotic expansion of solutions as domains become thinner.

Abstract

We are motivated by the study of the Microcanonical Variational Principle within the Onsager's description of two-dimensional turbulence in the range of energies where the equivalence of statistical ensembles fails. We obtain sufficient conditions for the existence and multiplicity of solutions for the corresponding Mean Field Equation on convex and "thin" enough domains in the supercritical (with respect to the Moser-Trudinger inequality) regime. This is a brand new achievement since existence results in the supercritical region were previously known \un{only} on multiply connected domains. Then we study the structure of these solutions by the analysis of their linearized problems and also obtain a new uniqueness result for solutions of the Mean Field Equation on thin domains whose energy is uniformly bounded from above. Finally we evaluate the asymptotic expansion of those solutions with respect to the thinning parameter and use it together with all the results obtained so far to solve the Microcanonical Variational Principle in a small range of supercritical energies where the entropy is eventually shown to be concave.