Blow-up analysis for some mean field equations involving probability measures from statistical hydrodynamics

Motivated by the mean field equations with probability measure derived by Sawada-Suzuki and by Neri in the context of the statistical mechanics description of two-dimensional turbulence, we study the semilinear elliptic equation with probability measure: {equation*} -\Delta v=\lambda\int_I V(\alpha,x,v)e^{\alpha v}\,\Pda -\frac{\lambda}{|\Omega|}\iint_{I\times\Om}V(\alpha,x,v)e^{\alpha v}\,\Pda dx, {equation*} defined on a compact Riemannian surface. This equation includes the above mentioned equations of physical interest as special cases. For such an equation we study the blow-up properties of solution sequences. The optimal Trudinger-Moser inequality is also considered.