Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence

Abstract

We consider the mean field equation arising in the high-energy scaling limit of point vortices with a general circulation constraint, when the circulation number density is subject to a probability measure. Mathematically, such an equation is a non-local elliptic equation containing an exponential nonlinearity which depends on this probability measure. We analyze the behavior of blow-up sequences of solutions in relation to the circulation numbers. As an application of our analysis we derive an improved Trudinger-Moser inequality for the associated variational functional.

Minimal blow-up masses and existence of solutions for an asymmetric sinh-Poisson equation

“…Here, we are concerned with the existence and blow‐up properties of for general values of . It is known that unbounded solution sequences for problem necessarily concentrate on a finite set . Our first aim in this note is to derive the optimal lower bounds for the blow‐up masses, see Theorem below.…”

A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects

“…In the asymmetric case of (1) there are by now just partial results, see for example [23,24,22,27,29]. We treat the problem (1) by following the ideas in [12] which is concerned for the equation (5) (this idea was first introduced in [18] for the SU (3) Toda system).…”

Blow-up analysis for Boltzmann–Poisson equation in Onsager's theory for point vortices with multi-intensities

“…Equation (2) is mathematically justified by the minimizing free energy method in the canonical formulation [2,9], and its mathematical analysis has revealed the quantized blow-up mechanism of sequences of solutions, see, e.g., [1,11,12,24,25,26]. Especially, the Y. Y. Li type estimate which is the behavior of blow-up solutions for (2) near the blow-up points has been studied [7,10].…”