## Differential and Integral Equations

- Differential Integral Equations
- Volume 25, Number 3/4 (2012), 201-222.

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

Tonia Ricciardi Ricciardi and Gabriella Zecca

#### Abstract

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: \begin{equation*} -\Delta v=\lambda\int_IV(\alpha,x,v)e^{\alpha v}\,{\mathcal P(d\alpha)} -\frac{\lambda}{|\Omega|}\iint_{I\times\Omega}V(\alpha,x,v)e^{\alpha v}\,{\mathcal P(d\alpha)} dx, \end{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.

#### Article information

**Source**

Differential Integral Equations, Volume 25, Number 3/4 (2012), 201-222.

**Dates**

First available in Project Euclid: 20 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356012734

**Mathematical Reviews number (MathSciNet)**

MR2917882

**Zentralblatt MATH identifier**

1265.76005

**Subjects**

Primary: 76B03: Existence, uniqueness, and regularity theory [See also 35Q35] 35B44: Blow-up 76B44

#### Citation

Ricciardi, Tonia Ricciardi; Zecca, Gabriella. Blow-up analysis for some mean field equations involving probability measures from statistical hydrodynamics. Differential Integral Equations 25 (2012), no. 3/4, 201--222. https://projecteuclid.org/euclid.die/1356012734