A Nonlinear Degenerate Advection-Diffusion Model for Crowd Instability and Stampede Risk Prediction

Crowd disasters arise from nonlinear interactions between density, spatial constraints, andlimited mobility. We propose a nonlinear advection-diffusion system for pedestrian densityρ(x, t) that captures three behavioral regimes: potential-driven flow at low density, diffusivespreading at moderate density, and nonlinear compression at high density. The model com-bines a harmonic potential field, density-dependent velocity, weak diffusion, and a smoothsuperquadratic pressure law. A risk functional R(x, t) is introduced to detect high-risk re-gions through rapid growth in density and velocity gradients. The framework provides acomputationally efficient alternative to agent-based simulations while retaining key instabil-ity mechanisms observed in real crowds.