Singularity formation for a fluid mechanics model with nonlocal velocity

TL;DR

This paper demonstrates that for a 1D fluid mechanics model with nonlocal velocity, solutions can lose regularity if initial data are not strictly positive, highlighting the importance of positivity for global regularity.

Contribution

It constructs initial data leading to loss of regularity, showing positivity is critical for the global regularity of the model and extends results to multi-dimensional cases.

Findings

Solutions can lose $C^1$ regularity with certain initial data.

Strict positivity of initial data is essential for global regularity.

Results extend to multi-dimensional models.

Abstract

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.