Luca Alasio (INRIA Paris, équipe Musclees): A family of PDEs for active particles with position and angle

We consider a family of macroscopic PDE models governing the space, time, and angle-dependent density of active particles in a periodic setting. Arising as the limit of stochastic lattice models, these degenerate parabolic equations are intrinsically non-local because their diffusive and drift coefficients depend on the angle-average of the solution. We establish that weak solutions become smooth for positive times under unrestrictive non-degeneracy conditions on the initial data. Under stronger initial constraints, we also prove the uniqueness of the weak solution, however uniqueness for irregular initial data remains unclear.

References:
Alasio, L., & Schulz, S. M. (2025). Regularity and uniqueness for a model of active particles with angle-averaged diffusions. Nonlinear Differential Equations and Applications NoDEA.

Alasio, L., Guerand, J., & Schulz, S. (2025). Regularity and trend to equilibrium for a non-local advection-diffusion model of active particles. Kinetic and Related Models.

Bruna, M., Burger, M., Esposito, A., & Schulz, S. M. (2022). Phase Separation in Systems of Interacting Active Brownian Particles. SIAM Journal on Applied Mathematics.