An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics: local well-posedness of paracontrolled solutions

Adrian Martini; Avi Mayorcas

doi:10.1007/s40072-024-00343-y

An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics: local well-posedness of paracontrolled solutions

Adrian Martini, Avi Mayorcas

Department of Mathematical Sciences

University of Oxford

Research output: Contribution to journal › Article › peer-review

Abstract

Using the method of paracontrolled distributions, we show the local well-posedness of an additive-noise approximation to the fluctuating hydrodynamics of the Keller–Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green’s function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.

Original language	English
Journal	Stochastics and Partial Differential Equations: Analysis and Computations
Early online date	24 Jan 2025
DOIs	https://doi.org/10.1007/s40072-024-00343-y
Publication status	E-pub ahead of print - 24 Jan 2025

Acknowledgements

We would like to express our gratitude to A. Etheridge, B. Fehrman, N. Perkowski and W. van Zuijlen for helpful discussions during the writing of this manuscript. We also wish to thank S. Mahdisoltani for bringing the subject of linear fluctuating hydrodynamics to our attention, in particular through the paper [40].

Funding

A. Martini was supported by the Engineering and Physical Sciences Research Council Doctoral Training Partnerships [Grant Number EP/R513295/1] and by the Lamb & Flag Scholarship of St John’s College, Oxford. Part of this work was completed during A. Martini’s participation in the Junior Trimester Program ‘Stochastic modelling in the life science: From evolution to medicine’ at the Hausdorff Research Institute for Mathematics, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy—EXC-2047/1—390685813. Work on this paper was undertaken during A. Mayorcas’s tenure as INI-Simons Post Doctoral Research Fellow hosted by the Isaac Newton Institute for Mathematical Sciences (INI) participating in programme Frontiers in Kinetic Theory, and by the Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge. This author would like to thank INI and DPMMS for support and hospitality during this fellowship, which was supported by Simons Foundation (award ID 316017) and by Engineering and Physical Sciences Research Council (EPSRC) Grant Number EP/R014604/1.

Funders	Funder number
Engineering and Physical Sciences Research Council

Keywords

Dean–Kawasaki equation
Linear fluctuating hydrodynamics
Parabolic-elliptic Keller–Segel model
Paracontrolled distributions
Singular stochastic partial differential equation

ASJC Scopus subject areas

Statistics and Probability
Modelling and Simulation
Applied Mathematics

Access to Document

10.1007/s40072-024-00343-yLicence: CC BY

Cite this

@article{b8ab3f5c197a4bb6b1e75bc369a583e9,

title = "An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics: local well-posedness of paracontrolled solutions",

abstract = "Using the method of paracontrolled distributions, we show the local well-posedness of an additive-noise approximation to the fluctuating hydrodynamics of the Keller–Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green{\textquoteright}s function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.",

keywords = "Dean–Kawasaki equation, Linear fluctuating hydrodynamics, Parabolic-elliptic Keller–Segel model, Paracontrolled distributions, Singular stochastic partial differential equation",

author = "Adrian Martini and Avi Mayorcas",

year = "2025",

month = jan,

day = "24",

doi = "10.1007/s40072-024-00343-y",

language = "English",

journal = "Stochastics and Partial Differential Equations: Analysis and Computations",

issn = "2194-0401",

publisher = "Springer New York",

}

TY - JOUR

T1 - An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics

T2 - local well-posedness of paracontrolled solutions

AU - Martini, Adrian

AU - Mayorcas, Avi

PY - 2025/1/24

Y1 - 2025/1/24

N2 - Using the method of paracontrolled distributions, we show the local well-posedness of an additive-noise approximation to the fluctuating hydrodynamics of the Keller–Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green’s function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.

AB - Using the method of paracontrolled distributions, we show the local well-posedness of an additive-noise approximation to the fluctuating hydrodynamics of the Keller–Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green’s function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.

KW - Dean–Kawasaki equation

KW - Linear fluctuating hydrodynamics

KW - Parabolic-elliptic Keller–Segel model

KW - Paracontrolled distributions

KW - Singular stochastic partial differential equation

UR - http://www.scopus.com/inward/record.url?scp=85217218165&partnerID=8YFLogxK

U2 - 10.1007/s40072-024-00343-y

DO - 10.1007/s40072-024-00343-y

M3 - Article

AN - SCOPUS:85217218165

SN - 2194-0401

JO - Stochastics and Partial Differential Equations: Analysis and Computations

JF - Stochastics and Partial Differential Equations: Analysis and Computations

ER -

An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics: local well-posedness of paracontrolled solutions

Abstract

Acknowledgements

Funding

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this