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 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.

Keywords

porous medium flow, quasi-geostrophic equations, the Euler alignment equation, singularity formation

2010 Mathematics Subject Classification

35Q35, 35Q92

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This work is supported by NSF grant DMS 1853001.

Received 30 August 2017

Accepted 27 March 2019

Published 6 January 2020