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

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