Blow-up analysis for Neri's mean field equation in 2D-turbulence

Abstract

In this paper we study the blow-up analysis for some mean field equation on point vortices which is derived by C. Neri under a stochastic assumption. In particular, we derive some estimate which is the asymptotic behavior of blow-up solutions near the blow-up points. To obtain this result we shall employ the new scaling argument for blow-up solutions of Neri's mean field equation. Moreover, we also study the power type elliptic equation in higher dimension case related to mean field equation and derive such estimate.

COMMUNICATED BY:

• R. Magnanini

Keywords:

• Blow-up analysis
• Nonlinear elliptic equation
• Mean field equation
• Point vortices

2010 Mathematics Subject Classifications:

• 35J61
• 39B52
• 76F99

Acknowledgments

The author would like to thank Professor Tonia Ricciardi from Università di Napoli Federico II, for valuable comments. This work was supported by the Center for Mathematical Modeling and Data Science, Osaka University and Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics).

Disclosure statement

No potential conflict of interest was reported by the author(s).