Active Crowds

Bruna, María; Burger, Martin; Pietschmann, Jan‐Frederik; Wolfram, Marie-Thérèse

doi:10.48550/arxiv.2107.06392

Cited by 1 publication

(1 citation statement)

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“…e.g. [5,8,12,13]). Denote by X i ∈ Ω ⊂ R 2 and Θ i ∈ T := R/(2πZ) the position and orientation of particle number i, respectively.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of an integro-differential model for active Brownian particles

Bruna¹,

Burger²,

Esposito³

et al. 2021

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We propose a general strategy for solving nonlinear integro-differential evolution problems with periodic boundary conditions, where no direct maximum/minimum principle is available. This is motivated by the study of recent macroscopic models for active Brownian particles with repulsive interactions, consisting of advection-diffusion processes in the space of particle position and orientation. We focus on one of such models, namely a semilinear parabolic equation with a nonlinear active drift term, whereby the velocity depends on the particle orientation and angle-independent overall particle density (leading to a nonlocal term by integrating out the angular variable). The main idea of the existence analysis is to exploit a-priori estimates from (approximate) entropy dissipation. The global existence and uniqueness of weak solutions is shown using a two-step Galerkin approximation with appropriate cutoff in order to obtain nonnegativity, an upper bound on the overall density and preserve a-priori estimates. Our anyalysis naturally includes the case of finite systems, corresponding to the case of a finite number of directions. The Duhamel principle is then used to obtain additional regularity of the solution, namely continuity in time-space. Motivated by the class of initial data relevant for the application, which includes perfectly aligned particles (same orientation), we extend the well-posedness result to very weak solutions allowing distributional initial data with low regularity.

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“…e.g. [5,8,12,13]). Denote by X i ∈ Ω ⊂ R 2 and Θ i ∈ T := R/(2πZ) the position and orientation of particle number i, respectively.…”

Section: Introductionmentioning

confidence: 99%