Abstract

The Dean-Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, and driven by space-time white noise; this equation describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Well-posedness for the Dean-Kawasaki model is open except for the simplest, purely diffusive, case, corresponding to overdamped Langevin dynamics. There, it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist. We derive and analyse a suitably regularised Dean-Kawasaki model of wave equation type driven by coloured noise, corresponding to second order Langevin dynamics, in one space dimension. The regularisation can be interpreted as considering particles of finite size rather than describing them by atomic measures. We establish existence and uniqueness of a solution. Specifically, we prove a high-probability result for the existence and uniqueness of mild solutions to this regularised Dean-Kawasaki model.
LanguageEnglish
JournalSiam Journal on Mathematical Analysis
Volume5
DOIs
StatusPublished - 10 Apr 2019

Keywords

  • math.PR
  • math-ph
  • math.AP
  • math.MP
  • 60H15, 35R60

Cite this

@article{f92a5b6ff87f4c4ead056768728a6684,
title = "A regularised Dean-Kawasaki model: derivation and analysis",
abstract = "The Dean-Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, and driven by space-time white noise; this equation describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Well-posedness for the Dean-Kawasaki model is open except for the simplest, purely diffusive, case, corresponding to overdamped Langevin dynamics. There, it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist. We derive and analyse a suitably regularised Dean-Kawasaki model of wave equation type driven by coloured noise, corresponding to second order Langevin dynamics, in one space dimension. The regularisation can be interpreted as considering particles of finite size rather than describing them by atomic measures. We establish existence and uniqueness of a solution. Specifically, we prove a high-probability result for the existence and uniqueness of mild solutions to this regularised Dean-Kawasaki model.",
keywords = "math.PR, math-ph, math.AP, math.MP, 60H15, 35R60",
author = "Federico Cornalba and Tony Shardlow and Johannes Zimmer",
note = "48 pages, 1 figure",
year = "2019",
month = "4",
day = "10",
doi = "10.1137/18M1172697",
language = "English",
volume = "5",
journal = "Siam Journal on Mathematical Analysis",
issn = "0036-1410",
publisher = "SIAM",

}

TY - JOUR

T1 - A regularised Dean-Kawasaki model

T2 - Siam Journal on Mathematical Analysis

AU - Cornalba, Federico

AU - Shardlow, Tony

AU - Zimmer, Johannes

N1 - 48 pages, 1 figure

PY - 2019/4/10

Y1 - 2019/4/10

N2 - The Dean-Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, and driven by space-time white noise; this equation describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Well-posedness for the Dean-Kawasaki model is open except for the simplest, purely diffusive, case, corresponding to overdamped Langevin dynamics. There, it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist. We derive and analyse a suitably regularised Dean-Kawasaki model of wave equation type driven by coloured noise, corresponding to second order Langevin dynamics, in one space dimension. The regularisation can be interpreted as considering particles of finite size rather than describing them by atomic measures. We establish existence and uniqueness of a solution. Specifically, we prove a high-probability result for the existence and uniqueness of mild solutions to this regularised Dean-Kawasaki model.

AB - The Dean-Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, and driven by space-time white noise; this equation describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Well-posedness for the Dean-Kawasaki model is open except for the simplest, purely diffusive, case, corresponding to overdamped Langevin dynamics. There, it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist. We derive and analyse a suitably regularised Dean-Kawasaki model of wave equation type driven by coloured noise, corresponding to second order Langevin dynamics, in one space dimension. The regularisation can be interpreted as considering particles of finite size rather than describing them by atomic measures. We establish existence and uniqueness of a solution. Specifically, we prove a high-probability result for the existence and uniqueness of mild solutions to this regularised Dean-Kawasaki model.

KW - math.PR

KW - math-ph

KW - math.AP

KW - math.MP

KW - 60H15, 35R60

U2 - 10.1137/18M1172697

DO - 10.1137/18M1172697

M3 - Article

VL - 5

JO - Siam Journal on Mathematical Analysis

JF - Siam Journal on Mathematical Analysis

SN - 0036-1410

ER -