2D Trudinger–Moser inequality for Boltzmann–Poisson equation with continuously distributed multi-intensities

Abstract

In this paper we study a functional associated with the mean filed limit of the point vortex distribution, that is,

$$\begin{aligned} J_{\lambda }(v)=\frac{1}{2}\Vert \nabla v\Vert _2^2-\lambda \int _{I_+}\log \Big (\int _{{\varOmega }} e^{\alpha v} dx \Big ){\mathcal {P}}(d\alpha ), \quad v \in H_0^1({\varOmega }) \end{aligned}$$

where $\lambda >0$ is a constant, ${\varOmega }\subset {\mathbb {R}}^2$ is a smooth bounded domain and ${\mathcal {P}}(d\alpha )$ is a Borel probability measure on $I_+=[0, 1]$. We show the boundedness of $J_{\lambda }$ from below, with borderline inequality in $\lambda$ when ${\mathcal {P}}(d\alpha )$ is continuous and satisfies the suitable assumptions.