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

  • Takashi Suzuki
  • Yohei ToyotaEmail author


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.

Mathematics Subject Classification

39B52 35J65 76F99 



The authors are grateful to the referee for careful reading of the manuscript and giving invaluable comments. This work is supported by JSPS Grant-in-Aid for Scientific Research (A) 26247013.


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Copyright information

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Authors and Affiliations

  1. 1.Center for Mathematical Modeling and Data ScienceOsaka UniversityOsakaJapan
  2. 2.Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering ScienceOsaka UniversityOsakaJapan

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