On a class of nonlocal continuity equations on graphs

Abstract

Motivated by applications in data science, we study partial differential equations on graphs. By a classical fixed-point argument, we show existence and uniqueness of solutions to a class of nonlocal continuity equations on graphs. We consider general interpolation functions, which give rise to a variety of different dynamics, for example, the nonlocal interaction dynamics coming from a solution-dependent velocity field. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen.

On evolution PDEs on co-evolving graphs

“…As we are not in the traditional Euclidean setting, we recall the notion of gradient and divergence for a function defined on graphs, similar to [20]; the only difference is that we do not restrict fluxes on the set G, as this is changing in time, but rather consider their (possible) extension to R 2d , cf. Remark 2.6.…”

“…In this work, we consider a class of nonlocal continuity equations on co-evolving graphs extending the results in [21,20], where the underlying graph is, instead, static. More precisely, in [21,20] the authors consider equations on the time interval [0, T ] of the form…”

“…The nonlinear heat equation on graphs was studied in [35,36], while a well-posedness theory for the generalised porous medium equation on infinite graphs can be found in [11]. In [21], the focus is on nonlocal dynamics on graphs using an upwind interpolation, while [20] concerns a class of continuity equations on graphs with general interpolations. The analysis of [21] is extended to two species with cross-interactions as well as nonlinear mobilities and α-homogeneous flux-velocity relations for α > 0 in [28,27].…”

Graph‐to‐local limit for a multi‐species nonlocal cross‐interaction system

“…Following refs. [1,3,4], our model is set on an abstract graph, where vertices are represented by a Radon measure 𝜇 ∈  + (ℝ 𝑑 ), which we call the base measure. Intuitively, the support of 𝜇 defines the underlying set of vertices, that is, 𝑉 = supp 𝜇.…”

“…We refer to refs. [1,3,4] for more details. Another important aspect is to deal with a large number of entities, for instance individuals or data; hence it is crucial to consider discrete and continuum models.…”