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arXiv · 2012.01966

Deterministic particle approximation of aggregation-diffusion equations on unbounded domains

Sara Daneri·Emanuela Radici·Eris Runa
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Abstract

We consider a one-dimensional aggregation-diffusion equation, which is the gradient flow in the Wasserstein space of a functional with competing attractive-repulsive interactions. We prove that the fully deterministic particle approximations with piecewise constant densities introduced in~\cite{Di Francesco-Rosini} starting from general bounded initial densities converge strongly in $L^1$ to bounded weak solutions of the PDE. In particular, the result is achieved in unbounded domains and for arbitrary nonnegative bounded initial densities, thus extending the results in \cite{Gosse-Toscani, Matthes-Osberger, Mathes-Soellner} (in which a no-vacuum condition is required) and giving an alternative approach to \cite{Carrillo-Craig-Patacchini} in the one-dimensional case, including also subquadratic and superquadratic diffusions.

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Sara Daneri, Emanuela Radici, Eris Runa. 2020-12-03. Deterministic particle approximation of aggregation-diffusion equations on unbounded domains. https://arxiv.org/abs/2012.01966

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