4 Citations (SciVal)


A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension d∈N. It is a regularised and inertial version of the Dean–Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in d-dimensions when the particle-width ϵ is proportional to N−1/θ for θ>2d and N is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the d-spatial dimensions, and use of the Faà di Bruno's formula.

Original languageEnglish
Pages (from-to)253-283
Number of pages31
JournalJournal of Differential Equations
Early online date5 Mar 2021
Publication statusPublished - 25 May 2021

Bibliographical note

Funding Information:
All authors thank the anonymous referee for his/her careful reading of the manuscript and valuable suggestions. This paper was motivated by stimulating discussions at the First Berlin–Leipzig Workshop on Fluctuating Hydrodynamics in August 2019 with Ana Djurdjevac, Rupert Klein and Ralf Kornhuber. JZ gratefully acknowledges funding by a Royal Society Wolfson Research Merit Award . FC gratefully acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754411.

Publisher Copyright:
© 2021 The Author(s)


  • Bessel functions of first kind
  • Fractional Sobolev spaces
  • Multi-dimensional Dean–Kawasaki model
  • Spatial regularisation
  • von Mises kernel
  • Well-posedness of stochastic PDEs

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics
  • Statistics and Probability


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