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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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).
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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