Well-posedness and stationary states for a crowded active Brownian system with size-exclusion

Martin Burger; Simon M. Schulz

doi:10.3934/dcds.2025006

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2025, Volume 45, Issue 8: 2882-2915. Doi: 10.3934/dcds.2025006

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Well-posedness and stationary states for a crowded active Brownian system with size-exclusion

Martin Burger^1, and
Simon M. Schulz^2, ,

1.
Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany
2.
Scuola Normale Superiore, CRM De Giorgi, Piazza dei Cavalieri, 3, 56126 Pisa, Italy

^*Corresponding author: Simon M. Schulz
^*Corresponding author: Simon M. Schulz

Received: July 2024

Revised: December 2024

Early access: January 2025

Published: August 2025

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Abstract

We prove the existence of solutions to a non-linear, non-local, degenerate equation which was previously derived as the formal hydrodynamic limit of an active Brownian particle system, where the particles are endowed with a position and an orientation. This equation incorporates diffusion in both the spatial and angular coordinates, as well as a non-linear non-local drift term, which depends on the angle-independent density. The spatial diffusion is non-linear degenerate and also comprises diffusion of the angle-independent density, which one may interpret as cross-diffusion with infinitely many species. Our proof relies on interpreting the equation as the perturbation of a gradient flow in a Wasserstein-type space. It generalizes the boundedness-by-entropy method to this setting and makes use of a gain of integrability due to the angular diffusion. For this latter step, we adapt a classical interpolation lemma for function spaces depending on time. We also prove uniqueness in the particular case where the non-local drift term is null, and provide existence and uniqueness results for stationary equilibrium solutions.

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Mathematics Subject Classification: Primary: 35K65, 35K55; Secondary: 76M30, 35Q92.

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References

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