Dear Colleagues,

We would like to invite you to the following SPASS seminar, jointly organized by UniPi, SNS, UniFi and UniSi (abstract below):

*Well-posedness and stationary states for a crowded active Brownian system with size-exclusion* by *Simon Michaël Schulz* (Scuola Normale Superiore)

The seminar will take place on *TUE, 24.10.2023* at *14:00 CET *in Aula Seminari, Dipartimento di Matematica, University of Pisa and streamed online at the link below.

The organizers, A. Agazzi, G. Bet, A. Caraceni, F. Grotto, G. Zanco https://sites.google.com/unipi.it/spass https://www.google.com/url?q=https://sites.google.com/unipi.it/spass&source=gmail-imap&ust=1665669490000000&usg=AOvVaw07o0tKOUGmZjDDF2Ta7pQY

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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. 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. This is joint work with Martin Burger.