### Abstract

The Dean–Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, 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 specific diffusive cases, corresponding to overdamped Langevin dynamics. It was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (nonatomic) solutions exist. We derive and analyze a suitably regularized Dean–Kawasaki model of wave equation type driven by colored noise, corresponding to second-order Langevin dynamics, in one space dimension. The regularization 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 regularized Dean–Kawasaki model.

Language | English |
---|---|

Pages | 1137-1187 |

Number of pages | 51 |

Journal | Siam Journal on Mathematical Analysis |

Volume | 51 |

Issue number | 2 |

DOIs | |

Status | Published - 9 Apr 2019 |

### Keywords

- Dean–Kawasaki model
- Langevin dynamics
- Mild solutions
- Spatial regularization of space-time white noise
- Stochastic wave equation

### ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

**A regularised Dean-Kawasaki model : derivation and analysis.** / Cornalba, Federico; Shardlow, Tony; Zimmer, Johannes.

Research output: Contribution to journal › Article

*Siam Journal on Mathematical Analysis*, vol. 51, no. 2, pp. 1137-1187. https://doi.org/10.1137/18M1172697

}

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 - 2019/4/9

Y1 - 2019/4/9

N2 - The Dean–Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, 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 specific diffusive cases, corresponding to overdamped Langevin dynamics. It was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (nonatomic) solutions exist. We derive and analyze a suitably regularized Dean–Kawasaki model of wave equation type driven by colored noise, corresponding to second-order Langevin dynamics, in one space dimension. The regularization 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 regularized 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, 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 specific diffusive cases, corresponding to overdamped Langevin dynamics. It was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (nonatomic) solutions exist. We derive and analyze a suitably regularized Dean–Kawasaki model of wave equation type driven by colored noise, corresponding to second-order Langevin dynamics, in one space dimension. The regularization 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 regularized Dean–Kawasaki model.

KW - Dean–Kawasaki model

KW - Langevin dynamics

KW - Mild solutions

KW - Spatial regularization of space-time white noise

KW - Stochastic wave equation

UR - http://www.scopus.com/inward/record.url?scp=85065500081&partnerID=8YFLogxK

U2 - 10.1137/18M1172697

DO - 10.1137/18M1172697

M3 - Article

VL - 51

SP - 1137

EP - 1187

JO - Siam Journal on Mathematical Analysis

JF - Siam Journal on Mathematical Analysis

SN - 0036-1410

IS - 2

ER -