Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions

Tony Shardlow, Federico Cornalba, Johannes Zimmer

Research output: Contribution to journalArticlepeer-review


A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension . 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 for 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
JournalJournal of Differential Equations
Early online date5 Mar 2021
Publication statusE-pub ahead of print - 5 Mar 2021

Cite this