On strictly starshaped domains of second kind (see Definition 1.2) we find sufficient conditions which allow the solution of two long standing open problems closely related to the mean field equation (P_lambda) below. On one side we describe the global behaviour of the Entropy for the mean field Microcanonical Variational Principle ((MVP) for short) arising in the Onsager description of two-dimensional turbulence. This is the completion of well known results first established in . Among other things we find a full unbounded interval of strict convexity of the Entropy. On the other side, to achieve this goal, we have to provide a detailed qualitative description of the global branch of solutions of (P) emanating from lambda = 0 and crossing lambda = 8pi. This is the completion of well known results first established in  and  for 8, and it has an independent mathematical interest, since the shape of global branches of semilinear elliptic equations, with very few well known exceptions, are poorly understood. It turns out that the (MVP) suggests the right variable (which is the energy) to be used to obtain a global parametrization of solutions of (P_lambda). A crucial spectral simplification is obtained by using the fact that, by definition, solutions of the (MVP) maximize the entropy at fixed energy and total vorticity.
Bartolucci, D. (2019). GLOBAL BIFURCATION ANALYSIS OF MEAN FIELD EQUATIONS AND THE ONSAGER MICROCANONICAL DESCRIPTION OF TWO-DIMENSIONAL TURBULENCE. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 58, 18 [10.1007/s00526-018-1445-4].
|Tipologia:||Articolo su rivista|
|Citazione:||Bartolucci, D. (2019). GLOBAL BIFURCATION ANALYSIS OF MEAN FIELD EQUATIONS AND THE ONSAGER MICROCANONICAL DESCRIPTION OF TWO-DIMENSIONAL TURBULENCE. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 58, 18 [10.1007/s00526-018-1445-4].|
|Altre informazioni significative:||Research partially supported by FIRB project ”Analysis and Beyond”, by PRIN project 2012, ”Variational and perturbative aspects in nonlinear differential problems”, and by the Consolidate the Foundations project 2015 (sponsored by Univ. of Rome ”Tor Vergata”), ”Nonlinear Differential Problems and their Applications|
|IF:||Con Impact Factor ISI|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Revisione (peer review):||Esperti anonimi|
|Digital Object Identifier (DOI):||http://dx.doi.org/10.1007/s00526-018-1445-4|
|Stato di pubblicazione:||Pubblicato|
|Data di pubblicazione:||2019|
|Titolo:||GLOBAL BIFURCATION ANALYSIS OF MEAN FIELD EQUATIONS AND THE ONSAGER MICROCANONICAL DESCRIPTION OF TWO-DIMENSIONAL TURBULENCE|
|Appare nelle tipologie:||01 - Articolo su rivista|