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

José Antonio Carrillo; Marco Di Francesco; Antonio Esposito; Simone Fagioli; Markus Schmidtchen

doi:10.3934/dcds.2020075

Article Contents

2020, Volume 40, Issue 2: 1191-1231. Doi: 10.3934/dcds.2020075

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

1.
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
2.
DISIM - Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Via Vetoio 1 (Coppito) 67100 L'Aquila (AQ), Italy

^* Corresponding author: José A. Carrillo
^* Corresponding author: José A. Carrillo

Received: May 31, 2019

Revised: August 31, 2019

Published: November 2019

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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 Brézis. 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.

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Mathematics Subject Classification: Primary: 45K05, 35A15; Secondary: 92D25, 35L45, 35L80.

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Figure 1. This example has two separated indicator functions as initial data. In the left graph we see the evolution of system (6) to the stationary state (right graph). In the middle we see the energy (black) of the solution and its dissipation (red). The dotted line is the numerical time derivative of the energy. It matches well with the analytically obtained dissipation

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Figure 2. We choose partially overlapping initial data and observe, as before that mixing occurs. The graph on the left displays the evolution of both densities at different time instances, while the rightmost graph displays the stationary state with identical densities. The graph in the middle shows the energy decay along the solution and the numerical dissipation and the analytical dissipation agree well

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Figure 3. Initial (left) and exact solution (right) at time $ t = 0.5 $ for the case of two distinct Dirac deltas at the level of distribution functions

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Figure 4. Initial (left) and exact solution (right) at time $ t = 1/(4(1-m)) $ with $ m = 0.4 $ for the case of two partially overlapping deltas at the level of distribution functions

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