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Graph-to-local limit for the nonlocal interaction equation

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arxiv 2306.03475 v2 pith:J65RPM23 submitted 2023-06-06 math.AP cs.NAmath.NA

Antonio Esposito , Georg Heinze , André Schlichting This is my paper · ORCID
classification math.APcs.NAmath.NA
keywords nonlocalequationinteractionclassequationsgraphssolutionsspace
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We study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively. More precisely, we prove that weak solutions of the nonlocal interaction equation on graphs converge to weak solutions of the aforementioned class of nonlocal interaction equation with a tensor-mobility in the Euclidean space. This highlights an interesting property of the graph, being a potential space-discretisation for the equation under study.

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Cited by 3 Pith papers

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  1. Stochastic homogenisation of nonlinear minimum-cost flow problems

    math.AP 2024-12 accept novelty 7.0 of 10

    On stationary random graphs satisfying quantitative connectedness, nonlinear minimum-cost flow problems Gamma-converge under rescaling to a continuum divergence-constrained problem with a homogenised energy density fr...

  2. Nonlocal approximation of an anisotropic cross-diffusion system

    math.AP 2024-12 conditional novelty 6.0 of 10

    Weak solutions of an anisotropic nonlocal cross-diffusion system converge to weak solutions of the corresponding local cross-diffusion system in the vanishing viscosity limit.

  3. Gradient flows on metric graphs with reservoirs: Microscopic derivation and multiscale limits

    math.AP 2024-12 accept novelty 6.0 of 10

    Coupled edge-diffusion and vertex-reservoir equations on metric graphs are shown to be generalized gradient flows, with existence proven via a discrete jump-process approximation and three EDP scaling limits established.

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