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
- Hantaek Bae, Rafael Granero-Belinchón, and Omar Lazar, On the local and global existence of solutions to 1D transport equations with nonlocal velocity, Netw. Heterog. Media 14 (2019), no. 3, 471–487. MR 3985391, DOI 10.3934/nhm.2019019
- Hantaek Bae, Rafael Granero-Belinchón, and Omar Lazar, Global existence of weak solutions to dissipative transport equations with nonlocal velocity, Nonlinearity 31 (2018), no. 4, 1484–1515. MR 3816643, DOI 10.1088/1361-6544/aaa2e0
- Gregory R. Baker, Xiao Li, and Anne C. Morlet, Analytic structure of two $1$D-transport equations with nonlocal fluxes, Phys. D 91 (1996), no. 4, 349–375. MR 1382265, DOI 10.1016/0167-2789(95)00271-5
- A. P. Calderón and A. Zygmund, Singular integrals and periodic functions, Studia Math. 14 (1954), 249–271 (1955). MR 69310, DOI 10.4064/sm-14-2-249-271
- Peter Constantin, Andrei Tarfulea, and Vlad Vicol, Long time dynamics of forced critical SQG, Comm. Math. Phys. 335 (2015), no. 1, 93–141. MR 3314501, DOI 10.1007/s00220-014-2129-3
- Peter Constantin and Jiahong Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré C Anal. Non Linéaire 25 (2008), no. 6, 1103–1110. MR 2466323, DOI 10.1016/j.anihpc.2007.10.001
- Antonio Córdoba and Diego Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys. 249 (2004), no. 3, 511–528. MR 2084005, DOI 10.1007/s00220-004-1055-1
- Antonio Córdoba, Diego Córdoba, and Marco A. Fontelos, Formation of singularities for a transport equation with nonlocal velocity, Ann. of Math. (2) 162 (2005), no. 3, 1377–1389. MR 2179734, DOI 10.4007/annals.2005.162.1377
- Michele Coti Zelati and Vlad Vicol, On the global regularity for the supercritical SQG equation, Indiana Univ. Math. J. 65 (2016), no. 2, 535–552. MR 3498176, DOI 10.1512/iumj.2016.65.5807
- Tam Do, On a 1D transport equation with nonlocal velocity and supercritical dissipation, J. Differential Equations 256 (2014), no. 9, 3166–3178. MR 3171771, DOI 10.1016/j.jde.2014.01.037
- Hongjie Dong, Well-posedness for a transport equation with nonlocal velocity, J. Funct. Anal. 255 (2008), no. 11, 3070–3097. MR 2464570, DOI 10.1016/j.jfa.2008.08.005
- Ning Ju, Dissipative 2D quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions, Indiana Univ. Math. J. 56 (2007), no. 1, 187–206. MR 2305934, DOI 10.1512/iumj.2007.56.2851
- Alexander Kiselev, Nonlocal maximum principles for active scalars, Adv. Math. 227 (2011), no. 5, 1806–1826. MR 2803787, DOI 10.1016/j.aim.2011.03.019
- A. Kiselev, Regularity and blow up for active scalars, Math. Model. Nat. Phenom. 5 (2010), no. 4, 225–255. MR 2662457, DOI 10.1051/mmnp/20105410
- Dong Li and Jose Rodrigo, Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation, Adv. Math. 217 (2008), no. 6, 2563–2568. MR 2397459, DOI 10.1016/j.aim.2007.11.002
- Luis Silvestre and Vlad Vicol, On a transport equation with nonlocal drift, Trans. Amer. Math. Soc. 368 (2016), no. 9, 6159–6188. MR 3461030, DOI 10.1090/tran6651
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