Singularity formation for a fluid mechanics model with nonlocal velocity

We study a 1D fluid mechanics model with nonlocal velocity. The equation can be viewed as a fractional porous medium flow, a 1D model of the quasi-geostrophic equation, and also a special case of Euler-Alignment system. For strictly positive smooth initial data, global regularity has been proved by Do, Kiselev, Ryzhik and Tan. We construct a family of non-negative smooth initial data so that solution loses $C^1$ regularity. Our result indicates that strict positivity is a critical condition to ensure global regularity of the system. We also extend our construction to the corresponding models in multi-dimensions.