Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential

Kaib, Gunnar

doi:10.48550/arxiv.1604.07298

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2017

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“…Various results on the analysis of energy minimizers respectively stationary states (cf. [2,5,16,19,22,26,27,29,32,39,50,58,59]) and the gradient flow dynamics of the form (cf. [6,9,8,16,28,30,36,65])…”

Section: Introductionmentioning

confidence: 99%

On a cross-diffusion model for multiple species with nonlocal interaction and size exclusion

2017

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The aim of this paper is to study a PDE model for two diffusing species interacting by local size exclusion and global attraction. This leads to a nonlinear degenerate crossdiffusion system, for which we provide a global existence result. The analysis is motivated by the formulation of the system as a formal gradient flow for an appropriate energy functional consisting of entropic terms as well as quadratic nonlocal terms. Key ingredients are entropy dissipation methods as well as the recently developed boundedness by entropy principle. Moreover, we investigate phase separation effects inherent in the cross-diffusion model by an analytical and numerical study of minimizers of the energy functional and their asymptotics to a previously studied case as the diffusivity tends to zero. Finally we briefly discuss coarsening dynamics in the system, which can be observed in numerical results and is motivated by rewriting the PDEs as a system of nonlocal Cahn-Hilliard equations.arXiv:1609.05024v2 [math.AP]

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