Mass quantization and minimax solutions for Neri's mean field equation in 2D-turbulence

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:260

Article

Ricciardi, T.¹  Zecca, G.¹ 

[1] Univ Naples Federico II, Dipartimento Matemat & Applicaz R Caccioppoli, Via Cintia, I-80126 Naples, Italy

关键词: Mean field equation;    Exponential nonlinearity;    Mass quantization;    Mountain pass solution;   

DOI  :  10.1016/j.jde.2015.08.045

来源: Elsevier

【 摘 要 】

We study the mean field equation derived by Neri in the context of the statistical mechanics description of 2D-turbulence, under a stochastic assumption on the vortex circulations. The corresponding mathematical problem is a nonlocal semilinear elliptic equation with exponential type nonlinearity, containing a probability measure P E M ([-1, 1]) which describes the distribution of the vortex circulations. Unlike the more investigated deterministic version, we prove that Neri's equation may be viewed as a perturbation of the widely analyzed standard mean field equation, obtained by taking P = 81. In particular, in the physically relevant case where P is non-negatively supported and P({1)) > 0, we prove the mass quantization for blow-up sequences. We apply this result to construct minimax type solutions on bounded domains in 1112 and on compact 2-manifolds without boundary. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free