2025/01/20 by Luca Alasio, Alasio, Luca, Simon Schulz +1 · 2 citations
Mathematics · Computer Science · Physics and Astronomy · #Mathematical Biology Tumor Growth #Advanced Mathematical Modeling in Engineering #Micro and Nano Robotics
paper · pdf · doi:10.48550/arxiv.2501.11488
We study the regularity and uniqueness of weak solutions of a degenerate parabolic equation, arising as the limit of a stochastic lattice model of self-propelled particles. The angle-average of the solution appears as a coefficient in the diffusive and drift terms, making the equation nonlocal. We prove that, under unrestrictive non-degeneracy assumptions on the initial data, weak solutions are smooth for positive times. Our method rests on deriving a drift-diffusion equation for a particular function of the angle-averaged density and applying De Giorgi's method to show that the original equation is uniformly parabolic for positive times. We employ a Galerkin approximation to justify rigorously the passage from divergence to non-divergence form of the equation, which yields improved estimates by exploiting a cancellation. By imposing stronger constraints on the initial data, we prove the uniqueness of the weak solution, which relies on Duhamel's principle and gradient estimates for the periodic heat kernel to derive L^∞ estimates for the angle-averaged density.