## Global smoothness for a 1D supercritical transport model with nonlocal velocity

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- by Lucas C. F. Ferreira and Valter V. C. Moitinho PDF
- Proc. Amer. Math. Soc.
**148**(2020), 2981-2995 Request permission

## Abstract:

We are concerned with a nonlocal transport 1D-model with supercritical dissipation $\gamma \in (0,1)$ in which the velocity is coupled via the Hilbert transform, namely the so-called CCF model. This model arises as a lower dimensional model for the well-known 2D dissipative quasi-geostrophic equation and in connection with vortex-sheet problems. It is known that its solutions can blow up in finite time when $\gamma \in (0,1/2)$. On the other hand, as stated by Kiselev (2010), in the supercritical subrange $\gamma \in \lbrack 1/2,1)$ it is an open problem to know whether its solutions are globally regular. We show global existence of nonnegative $H^{3/2}$-strong solutions in a supercritical subrange (close to 1) that depends on the initial data norm. Then, for each arbitrary smooth nonnegative initial data, the model has a unique global smooth solution provided that $\gamma \in \lbrack \gamma _{1},1)$ where $\gamma _{1}$ depends on the $H^{3/2}$-initial data norm. Our approach is inspired by that of Coti Zelati and Vicol (IUMJ, 2016).## References

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## Additional Information

**Lucas C. F. Ferreira**- Affiliation: IMECC-Department of Mathematics, State University of Campinas (Unicamp), Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, Sao Paulo, Brazil
- MR Author ID: 795159
- Email: lcff@ime.unicamp.br
**Valter V. C. Moitinho**- Affiliation: IMECC-Department of Mathematics, State University of Campinas (Unicamp), Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, Sao Paulo, Brazil
- Email: valtermoitinho@live.com
- Received by editor(s): June 11, 2019
- Received by editor(s) in revised form: November 21, 2019
- Published electronically: March 17, 2020
- Additional Notes: The first author was supported by FAPESP and CNPq, Brazil.

The second author was supported by CAPES and CNPq, Brazil. - Communicated by: Ryan Hynd
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 2981-2995 - MSC (2010): Primary 35Q35, 35B65, 76D03
- DOI: https://doi.org/10.1090/proc/14984
- MathSciNet review: 4099785