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


Authors: Lucas C. F. Ferreira and Valter V. C. Moitinho
Journal: Proc. Amer. Math. Soc. 148 (2020), 2981-2995
MSC (2010): Primary 35Q35, 35B65, 76D03
DOI: https://doi.org/10.1090/proc/14984
Published electronically: March 17, 2020
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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).


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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
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

DOI: https://doi.org/10.1090/proc/14984
Keywords: 1D transport model, nonlocal velocity, Hilbert transform, global regularity, supercritical dissipation
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
Article copyright: © Copyright 2020 American Mathematical Society