Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions

Carrillo JA, Di Francesco M, Esposito A, Fagioli S, Schmidtchen M (2020)

Publication Language: English

Publication Status: In press

Publication Type: Journal article

Future Publication Type: Journal article

Publication year: 2020

Journal

Discrete and Continuous Dynamical Systems American Institute of Mathematical Sciences (AIMS)

DOI: 10.3934/dcds.2020075

Abstract

We prove global-in-time existence and uniqueness of measure solutions of a nonlocal interaction system of two species in one spatial dimension. For initial data including atomic parts we provide a notion of gradient-flow solutions in terms of the pseudo-inverses of the corresponding cumulative distribution functions, for which the system can be stated as a gradient flow on the Hilbert space $L^2(0,1)^2$ according to the classical theory by Br\'ezis. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that the scheme preserves finiteness of the $L^m$-norms for all $m\in [1,+\infty]$ and of the second moments. We then provide a characterisation of equilibria and prove that they are achieved (up to time subsequences) in the large time asymptotics. We conclude the paper constructing two examples of non-uniqueness of measure solutions emanating from the same (atomic) initial datum, showing that the notion of gradient flow solution is necessary to single out a unique measure solution.

Authors with CRIS profile

Antonio Esposito Lehrstuhl für Angewandte Mathematik (Modellierung und Numerik)

Involved external institutions

Sorbonne Université FR France (FR) Imperial College London / The Imperial College of Science, Technology and Medicine GB United Kingdom (GB) University of L'Aquila / Università degli Studi dell'Aquila IT Italy (IT)

How to cite

APA:

Carrillo, J.A., Di Francesco, M., Esposito, A., Fagioli, S., & Schmidtchen, M. (2020). Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions. Discrete and Continuous Dynamical Systems. https://dx.doi.org/10.3934/dcds.2020075

MLA:

Carrillo, José A., et al. "Measure solutions to a system of continuity equations driven by Newtonian nonlocal interactions." Discrete and Continuous Dynamical Systems (2020).

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