caa a22 4500 605515700 CHVBK 20210128100710.0 cr unu---uuuuu 210128e20150801xx s 000 0 eng 10.1007/s00205-014-0836-8 doi (NATIONALLICENCE)springer-10.1007/s00205-014-0836-8 Supercritical Mean Field Equations on Convex Domains and the Onsager's Statistical Description of Two-Dimensional Turbulence [Elektronische Daten] [Daniele Bartolucci, Francesca De Marchis] 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. Springer-Verlag Berlin Heidelberg, 2015 Bartolucci Daniele Department of Mathematics, University of Rome "Tor Vergata”, Via della ricerca scientifica n.1, 00133, Rome, Italy aut De Marchis Francesca Department of Mathematics, University of Rome "Tor Vergata”, Via della ricerca scientifica n.1, 00133, Rome, Italy aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 217/2(2015-08-01), 525-570 0003-9527 217:2<525 2015 217 205 https://doi.org/10.1007/s00205-014-0836-8 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-014-0836-8 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Bartolucci Daniele Department of Mathematics, University of Rome "Tor Vergata”, Via della ricerca scientifica n.1, 00133, Rome, Italy aut NATIONALLICENCE 700 1- De Marchis Francesca Department of Mathematics, University of Rome "Tor Vergata”, Via della ricerca scientifica n.1, 00133, Rome, Italy aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 217/2(2015-08-01), 525-570 0003-9527 217:2<525 2015 217 205