A mean field type flow, II: Existence and convergence

Jean-Baptiste Castéras

Vol. 276 (2015), No. 2, 321–345

DOI: 10.2140/pjm.2015.276.321

Abstract

This paper is the continuation of (Castéras 2015), in which we investigated a gradient flow related to the mean field type equation. First, we show that this flow exists for all time. Next, using the compactness result of Castéras (2015), we prove, under a suitable hypothesis on its energy, the convergence of the flow to a solution of the mean field type equation. We also get a divergence result if the energy of the initial data is largely negative.

Keywords

mean field equation, blow-up analysis, geometric flow

Mathematical Subject Classification 2010

Primary: 35B33, 35J20, 53C44, 58E20

Milestones

Received: 14 June 2014

Revised: 2 December 2014

Accepted: 9 January 2015

Published: 15 July 2015