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

Cornalba F, Shardlow T, Zimmer J. 2021. Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions. Journal of Differential Equations. 284(5), 253–283.

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Author
Cornalba, FedericoIST Austria; Shardlow, Tony; Zimmer, Johannes
Department
Abstract
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.
Publishing Year
Date Published
2021-03-05
Journal Title
Journal of Differential Equations
Acknowledgement
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.
Volume
284
Issue
5
Page
253-283
ISSN
eISSN
IST-REx-ID

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Cornalba F, Shardlow T, Zimmer J. Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions. Journal of Differential Equations. 2021;284(5):253-283. doi:10.1016/j.jde.2021.02.048
Cornalba, F., Shardlow, T., & Zimmer, J. (2021). Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions. Journal of Differential Equations. Elsevier. https://doi.org/10.1016/j.jde.2021.02.048
Cornalba, Federico, Tony Shardlow, and Johannes Zimmer. “Well-Posedness for a Regularised Inertial Dean–Kawasaki Model for Slender Particles in Several Space Dimensions.” Journal of Differential Equations. Elsevier, 2021. https://doi.org/10.1016/j.jde.2021.02.048.
F. Cornalba, T. Shardlow, and J. Zimmer, “Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions,” Journal of Differential Equations, vol. 284, no. 5. Elsevier, pp. 253–283, 2021.
Cornalba F, Shardlow T, Zimmer J. 2021. Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions. Journal of Differential Equations. 284(5), 253–283.
Cornalba, Federico, et al. “Well-Posedness for a Regularised Inertial Dean–Kawasaki Model for Slender Particles in Several Space Dimensions.” Journal of Differential Equations, vol. 284, no. 5, Elsevier, 2021, pp. 253–83, doi:10.1016/j.jde.2021.02.048.
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