Free Boundary Problems in PDEs and Particle Systems

Carinci, Gioia; Masi, Anna De; Giardinà, Cristian; Presutti, Errico

doi:10.1007/978-3-319-33370-0

Cited by 21 publications

(30 citation statements)

References 14 publications

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“…Upper bound for all k. Since ρ(·, δ) ρ δ,+ (·, δ) and since C + δ T * δ preserves order (see Chapter 5 of [4])…”

Section: 5mentioning

confidence: 99%

Non local branching Brownian motions with annihilation and free boundary problems

Masi¹,

Ferrari²,

Presutti³

et al. 2019

Electron. J. Probab.

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We study a system of branching Brownian motions on R with annihilation: at each branching time a new particle is created and the leftmost one is deleted. In [7] it has been studied the case of strictly local creations (the new particle is put exactly at the same position of the branching particle), in [10] instead the position y of the new particle has a distribution p(x, y)dy, x the position of the branching particle, however particles in between branching times do not move. In this paper we consider Brownian motions as in [7] and non local branching as in [10] and prove convergence in the continuum limit (when the number N of particles diverges) to a limit density which satisfies a free boundary problem when this has classical solutions, local in time existence of classical solution has been proved recently in [13]. We use in the convergence a stronger topology than in [7] and [10] and have explicit bounds on the rate of convergence.

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“…Upper bound for all k. Since ρ(·, δ) ρ δ,+ (·, δ) and since C + δ T * δ preserves order (see Chapter 5 of [4])…”

Section: 5mentioning

confidence: 99%

Non local branching Brownian motions with annihilation and free boundary problems

Masi¹,

Ferrari²,

Presutti³

et al. 2019

Electron. J. Probab.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Typically in these models, all particles undergo motion, branching or nonlocal duplication, whereas the boundary particles are in addition subject to a removal (or injection) mechanism. The study of their hydrodynamic limits and the characterization of these limits in terms of free boundary problems (FBP) has been the subject of interest in the literature [4,5,6,7,8,10]. A prototype of this class of models is the following variant of the basic model studied by Carinci, De Masi, Giardinà and Presutti in the monograph [5].…”

Section: Introductionmentioning

confidence: 99%

“…The formulation is used to obtain new hydrodynamic limit results for two models. One is a variant of the main model studied by Carinci, De Masi, Giardinà and Presutti [5] where Brownian particles undergo injection according to a general injection measure, and removal that is restricted to the rightmost particle of the configuration. This partially addresses a conjecture of [5].…”

mentioning

confidence: 99%

“…One is a variant of the main model studied by Carinci, De Masi, Giardinà and Presutti [5] where Brownian particles undergo injection according to a general injection measure, and removal that is restricted to the rightmost particle of the configuration. This partially addresses a conjecture of [5]. Next a Brownian particle system is considered where the Q-quantile member of the population is removed until extinction, where Q is a given [0, 1]-valued continuous function of time.…”

mentioning

confidence: 99%

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The heat equation with order-respecting absorption and particle systems with topological interaction

Atar¹

2020

Preprint

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A PDE formulation is proposed, referred to as a heat equation with order-respecting absorption, aimed at characterizing hydrodynamic limits of a class of particle systems on the line with topological interaction that have so far been described by free boundary problems. It consists of the heat equation with measure-valued injection and absorption terms, where the absorption measure respects the usual order on R in the sense that, for all r ∈ R, it charges (−∞, r) only at times when the solution vanishes on (r, ∞). The formulation is used to obtain new hydrodynamic limit results for two models. One is a variant of the main model studied by Carinci, De Masi, Giardinà and Presutti [5] where Brownian particles undergo injection according to a general injection measure, and removal that is restricted to the rightmost particle of the configuration. This partially addresses a conjecture of [5]. Next a Brownian particle system is considered where the Q-quantile member of the population is removed until extinction, where Q is a given [0, 1]-valued continuous function of time. Here, unlike in earlier work on the subject, the removal mechanism acts on particles that are 'at the boundary' but are not rightmost or leftmost. Finally, further potential uses of order-respecting absorption are mentioned.

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“…In one dimension, motion by curvature has been proved for a number of interacting particle systems. It usually boils down to the heat equation in this case, and the Lifshitz law is related to freezing/melting problems, see [CS + 96][CK08] [CKG12], as well as [Lac14] and the monograph [CDMGP16]. In two dimensions, a landmark is the proof of anisotropic motion by curvature for the zero temperature Ising model with Glauber dynamics (or zero-temperature stochastic Ising model).…”

Section: Introductionmentioning

confidence: 99%

Motion by curvature and large deviations for an interface dynamics on Z 2

Dagallier

2020

Preprint

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We study large deviations for a Markov process on curves in Z 2 mimicking the motion of an interface. Our dynamics can be tuned with a parameter β, which plays the role of an inverse temperature, and coincides at β = ∞ with the zero-temperature Ising model with Glauber dynamics, where curves correspond to the boundaries of droplets of one phase immersed in a sea of the other one. We prove that contours typically follow a motion by curvature with an influence of the parameter β, and establish large deviations bounds at all large enough β < ∞. The diffusion coefficient and mobility of the model are identified and correspond to those predicted in the literature.

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Free Boundary Problems in PDEs and Particle Systems

Cited by 21 publications

References 14 publications

Non local branching Brownian motions with annihilation and free boundary problems

Non local branching Brownian motions with annihilation and free boundary problems

The heat equation with order-respecting absorption and particle systems with topological interaction

Motion by curvature and large deviations for an interface dynamics on Z 2

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