Mean field equations with probability measure in 2D-turbulence

Abstract

We outline some results obtained in recent years concerning a class of semilinear elliptic equations of the “mean field” type containing a probability measure, motivated by the statistical mechanics description of 2D turbulence. In particular, we exhibit the optimal Moser-Trudinger constants, we study the blow-up of solution sequences, we prove their mass quantization under suitable conditions and we consequently derive the existence of solutions by variational methods.