Supercritical Mean Field Equations on Convex Domains and the Onsager’s Statistical Description of Two-Dimensional Turbulence

Abstract

We are motivated by the study of the Microcanonical Variational Principle within 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 only on multiply connected domains. We then study the structure of these solutions by the analysis of their linearized problems and we 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, combining it with all the results obtained so far, we solve the Microcanonical Variational Principle in a small range of supercritical energies where the entropy is shown to be concave

On the supercritical mean field equation on pierced domains

“…Moreover, for D simply connected and λ = 8π, Chang, Chen and Lin [5] gave necessary and sufficient conditions for (1.1) to have a solution. To our knowledge, a part from a recent result on some convex domains obtained by Bartolucci and De Marchis [1] indicated there are no further results concerning the resonant supercritical case λ ∈ 8πN and λ > 8π. This result has been recently extended for not simply connected domains by Bartolucci and Lin [2].…”

“…There exist p 0 > 1 and 0 > 0 such that for any p ∈ (1, p 0 ), ∈ (0, 0 ) and R > 0 we have for any φ, φ 1 There exist p 0 > 1 and 0 > 0 such that for any p ∈ (1, p 0 ), ∈ (0, 0 ) and R > 0 we have for any φ, φ 1 …”

Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions

“…To simplify our notation, we shall always assume |M | = 1. Equation (1.3) and its counterpart on bounded planar domains arise in several areas of mathematics and physics and there are by now many results concerning existence ( [2,15,9,10,11,24,29,30,41,42]), uniqueness of solutions ( [4,12,13,14,26,47,48,49,60,65]) and blow-up analysis ( [3,5,16,17,19,28,31,57,58]). On one hand, they are derived as a mean field limit in the statistical mechanics description of two dimensional turbulent Euler flows ( [20,21]) and selfgravitating systems ( [54,56,72]).…”

Non-degeneracy and uniqueness of solutions to singular mean field equations on bounded domains

“…Definition 1.3. Let h = e H and let µ + be defined as in (6). Let ω ⊆ Ω be a nonempty subdomain and let ω be given as in Definition 1.2.…”

“…The strategy to prove Theorem 1.2 is inspired by the one in [23], with several non-trivial improvements needed to deal with general singular data (given by some measures as in (6)) and non-smooth domains. To this end, the first tool we need is an Alexandrov-Bol's inequality for solutions of (5) suitable for our setting.…”