2016/09/14 by Daniele Bartolucci, Bartolucci, Daniele · 1 citation
Mathematics · Engineering · Economics, Econometrics and Finance · #Navier-Stokes equation solutions #Fluid Dynamics and Turbulent Flows #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1609.04139
On strictly starshaped domains of second kind we find natural sufficient\nconditions which allow the solution of two long standing open problems closely\nrelated to the mean field equation prl below. On one side we catch the\nglobal behaviour of the Entropy for the mean field Microcanonical Variational\nPrinciple ((MVP) for short) arising in the Onsager description of\ntwo-dimensional turbulence. This is the completion of well known results first\nestablished in Caglioti et al. Comm.Math.Phys. (1995). Among other things we\nfind a full unbounded interval of strict convexity of the Entropy. On the other\nside, to achieve this goal, we have to provide a detailed qualitative\ndescription of the global branch of solutions of prl emanating from lm=0\nand crossing lm=8\π. This is the completion of well known results first\nestablished in Suzuki A.I.H.P. (1992) and Chang et al. New Stud. Adv. Math.\n(2003) for lm\≤ 8\π, and it has an independent mathematical interest,\nsince global branches of semilinear elliptic equations, with very few well\nknown exceptions, are poorly understood. The (MVP) suggests the right variable\n(which is the energy) to be used to obtain a global parametrization of\nsolutions of prl. A crucial spectral simplification is obtained by using the\nfact that, by definition, solutions of the (MVP) maximize the entropy at fixed\nenergy and total vorticity.\n