Well-posedness of the Dean--Kawasaki and the nonlinear Dawson--Watanabe equation with correlated noise

Fehrman, Benjamin; Gess, Benjamin

doi:10.48550/arxiv.2108.08858

Cited by 6 publications

(14 citation statements)

References 68 publications

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“…and linked it to the large deviation principle for such process in a suitable thermodynamic setting. A corresponding well-posedness result for truncated (low spatial frequency) noise and regularised nonlinearity has been obtained by Fehrman and Gess [12], see also [13].…”

Section: Introductionmentioning

confidence: 67%

The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

Cornalba¹,

Fischer²

2021

Preprint

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The Dean-Kawasaki equation -a strongly singular SPDE -is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers N1. The singular nature of the Dean-Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions.In the present work, we give a rigorous and fully quantitative justification of the Dean-Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean-Kawasaki equation may approximate the density fluctuations of N noninteracting diffusing particles to arbitrary order in N −1 (in suitable weak metrics). In other words, the Dean-Kawasaki equation may be interpreted as a "recipe" for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.

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Section: Introductionmentioning

confidence: 67%

The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

Cornalba¹,

Fischer²

2021

Preprint

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“…The definition of a kinetic solution given in Definition 2.5 includes only the conditions required to prove the regularity results in Theorem 1.2 and Theorem 1.3. Additional assumptions on the kinetic measure are needed in order to prove uniqueness of solutions [12,16]. Under these additional assumptions, the well-posedness of nonnegative kinetic solutions of (1) for the case α = 1 2 in Assumption 1.1 has been proven in [12] for locally 1 2 -Hölder continuous g k .…”

Section: Kinetic Solutionmentioning

confidence: 99%

“…Additional assumptions on the kinetic measure are needed in order to prove uniqueness of solutions [12,16]. Under these additional assumptions, the well-posedness of nonnegative kinetic solutions of (1) for the case α = 1 2 in Assumption 1.1 has been proven in [12] for locally 1 2 -Hölder continuous g k . The existence and uniqueness for signed kinetic solutions of (1) with the same type of g k is still an open problem.…”

Section: Kinetic Solutionmentioning

confidence: 99%

“…Remark 3.2. We introduce two cut-off functions in the time and velocity variable in the kinetic formulation (12) to prove Theorem 1.2 and Theorem 1.3 in Section 4. The Definition 3.1 is a modification of Definition 2.5 to take into account the presence of these cut-off functions.…”

Section: Averaging Lemmatamentioning

confidence: 99%

“…and let Ψ κ (v) = v 0 ψ κ (ṽ)dṽ. We may choose ϕ(t, x, v) = ζ l (t)ψ κ (v) and take the expectation in (12) to have…”

Section: Application To Stochastic Porous Medium Equationsmentioning

confidence: 99%

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Bruno,

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Weber

2021

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We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in space. The coefficients are assumed to be Hölder continuous and the cases of smooth coefficients of at most linear growth as well as √ u are covered by our assumptions. The regularity obtained is consistent with the optimal regularity derived for the deterministic porous medium equation in [15,18] and the presence of the temporal white noise. The proof relies on a significant adaptation of velocity averaging techniques from their usual L 1 context to the natural L 2 setting of the stochastic case. We introduce a new mixed kinetic/mild representation of solutions to quasilinear SPDE and use L 2 based a priori bounds to treat the stochastic term.

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Perspective: New directions in dynamical density functional theory

Vrugt

Wittkowski

2022

J. Phys.: Condens. Matter

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Classical dynamical density functional theory (DDFT) has become one of the central modeling approaches in nonequilibrium soft matter physics. Recent years have seen the emergence of novel and interesting fields of application for DDFT. In particular, there has been a remarkable growth in the amount of work related to chemistry. Moreover, DDFT has stimulated research on other theories such as phase field crystal models and power functional theory. In this perspective, we summarize the latest developments in the field of DDFT and discuss a variety of possible directions for future research.

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Well-posedness of the Dean--Kawasaki and the nonlinear Dawson--Watanabe equation with correlated noise

Cited by 6 publications

References 68 publications

The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles

Optimal regularity in time and space for stochastic porous medium equations

Perspective: New directions in dynamical density functional theory

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