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## Singularity formation for a fluid mechanics model with nonlocal velocity

Changhui Tan

Communications in Mathematical Sciences, Volume 17, No 7, pp. 1779-1794 (2019).

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 quasi-geostrophic equation, and also a special case of the Euler alignment system. For strictly positive smooth initial data, global regularity has been proved in [Do, Kiselev, Ryzhik and Tan, Arch. Ration. Mech. Anal., 228(1):1–37, 2018]. We construct a family of non-negative smooth initial data so that solution is not $$C^1$$-uniformly bounded. 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.

 doi:10.4310/CMS.2019.v17.n7.a2 Download the Published Version This work is supported by NSF grant DMS #1853001
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