Abstract
Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with $L^2$ initial data and minimal assumptions on the drift are locally Hölder continuous. As an application we show that solutions of the quasi-geostrophic equation with initial $L^2$ data and critical diffusion $(-\Delta)^{1/2}$ are locally smooth for any space dimension.
-
[Cordoba^2]
A. Córdoba, D. Córdoba, and M. A. Fontelos, "Formation of singularities for a transport equation with nonlocal velocity," Ann. of Math., vol. 162, iss. 3, pp. 1377-1389, 2005.
@article {Cordoba^2,
author = {C{ó}rdoba, Antonio and C{ó}rdoba, Diego and Fontelos, Marco A.},
TITLE = {Formation of singularities for a transport equation with nonlocal velocity},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {162},
YEAR = {2005},
NUMBER = {3},
PAGES = {1377--1389},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {35A20 (35Q35 82C70)},
MRNUMBER = {2179734},
DOI = {10.4007/annals.2005.162.1377},
ZBLNUMBER = {1101.35052},
} -
[Duvaut_Lions] G. Duvaut and J. -L. Lions, Les inéquations en mécanique et en physique, Paris: Dunod, 1972.
@book {Duvaut_Lions,
author = {Duvaut, G. and Lions, J.-L.},
TITLE = {Les inéquations en mécanique et en physique},
NOTE = {Travaux et Recherches Mathématiques, No. 21},
PUBLISHER = {Dunod},
ADDRESS = {Paris},
YEAR = {1972},
PAGES = {xx+387},
MRCLASS = {73.49},
MRNUMBER = {57 \#4778},
ZBLNUMBER = {0298.73001},
} -
[Cordoba^2_2004]
A. Córdoba and D. Córdoba, "A maximum principle applied to quasi-geostrophic equations," Comm. Math. Phys., vol. 249, iss. 3, pp. 511-528, 2004.
@article {Cordoba^2_2004,
author = {C{ó}rdoba, Antonio and C{ó}rdoba, Diego},
TITLE = {A maximum principle applied to quasi-geostrophic equations},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {249},
YEAR = {2004},
NUMBER = {3},
PAGES = {511--528},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {76B03 (26A33 35Q35 76U05 86A05)},
MRNUMBER = {2005f:76011},
MRREVIEWER = {Luigi Carlo Berselli},
DOI = {10.1007/s00220-004-1055-1},
ZBLNUMBER = {02158321},
} -
[Constantin_Lecture] P. Constantin, "Euler equations, Navier-Stokes equations and turbulence," in Mathematical foundation of turbulent viscous flows, New York: Springer-Verlag, 2006, pp. 1-43.
@incollection {Constantin_Lecture,
author = {Constantin, Peter},
TITLE = {Euler equations, {N}avier-{S}tokes equations and turbulence},
BOOKTITLE = {Mathematical foundation of turbulent viscous flows},
SERIES = {Lecture Notes in Math.},
NUMBER = {1871},
PAGES = {1--43},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2006},
MRCLASS = {76-02 (35Q35 37N10 76B03 76D03 76D05 76F02)},
MRNUMBER = {2196360},
ZBLNUMBER = {05551754},
} -
[Constantin_Cordoba_Wu] P. Constantin, D. Cordoba, and J. Wu, "On the critical dissipative quasi-geostrophic equation," Indiana Univ. Math. J., vol. 50, iss. Special Issue, pp. 97-107, 2001.
@article {Constantin_Cordoba_Wu,
author = {Constantin, Peter and Cordoba, Diego and Wu, Jiahong},
TITLE = {On the critical dissipative quasi-geostrophic equation},
NOTE = {Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000)},
JOURNAL = {Indiana Univ. Math. J.},
FJOURNAL = {Indiana University Mathematics Journal},
VOLUME = {50},
YEAR = {2001},
NUMBER = {Special Issue},
PAGES = {97--107},
ISSN = {0022-2518},
CODEN = {IUMJAB},
MRCLASS = {35Q35 (76B03 86A05)},
MRNUMBER = {2002h:35246},
MRREVIEWER = {Emmanuel Grenier},
ZBLNUMBER = {0989.86004},
} -
[Constantin_Wu] P. Constantin and J. Wu, "Behavior of solutions to 2D quasi-geostrophic equations," SIAM J. Math. Anal., vol. 30, pp. 937-948, 1999.
@article {Constantin_Wu,
author = {Constantin, Peter and Wu, Jiahong},
TITLE = {Behavior of solutions to 2D quasi-geostrophic equations},
JOURNAL = {{\rm SIAM} J. Math. Anal.},
VOLUME = {30},
YEAR = {1999},
PAGES = {937--948},
ZBLNUMBER = {0957.76093},
} -
[Chae2006]
D. Chae, "On the regularity conditions for the dissipative quasi-geostrophic equations," SIAM J. Math. Anal., vol. 37, iss. 5, pp. 1649-1656, 2006.
@article {Chae2006,
author = {Chae, Dongho},
TITLE = {On the regularity conditions for the dissipative quasi-geostrophic equations},
JOURNAL = {{\rm SIAM} J. Math. Anal.},
FJOURNAL = {SIAM Journal on Mathematical Analysis},
VOLUME = {37},
YEAR = {2006},
NUMBER = {5},
PAGES = {1649--1656},
ISSN = {0036-1410},
MRCLASS = {76B03 (35B45 35Q35 76D03 86Axx)},
MRNUMBER = {2215601},
DOI = {10.1137/040616954},
ZBLNUMBER = {1141.76010},
} -
[Chae_Lee] D. Chae and J. Lee, "Global well-posedness in the super-critical dissipative quasi-geostrophic equations," Comm. Math. Phys., vol. 233, iss. 2, pp. 297-311, 2003.
@article {Chae_Lee,
author = {Chae, Dongho and Lee, Jihoon},
TITLE = {Global well-posedness in the super-critical dissipative quasi-geostrophic equations},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {233},
YEAR = {2003},
NUMBER = {2},
PAGES = {297--311},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {76D03 (35B30 35Q35 76U05 86A05)},
MRNUMBER = {2004k:76031},
MRREVIEWER = {Isabelle Gallagher},
ZBLNUMBER = {1019.86002},
} -
[Cordoba_Blow-up]
D. Cordoba, "Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation," Ann. of Math., vol. 148, iss. 3, pp. 1135-1152, 1998.
@article {Cordoba_Blow-up,
author = {Cordoba, Diego},
TITLE = {Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {148},
YEAR = {1998},
NUMBER = {3},
PAGES = {1135--1152},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {76B03 (34C60 35Q35 76B99)},
MRNUMBER = {2000j:76020},
DOI = {10.2307/121037},
ZBLNUMBER = {0920.35109},
} -
[Wu_Besov_small]
J. Wu, "Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces," SIAM J. Math. Anal., vol. 36, iss. 3, pp. 1014-1030, 2004/05.
@article {Wu_Besov_small,
author = {Wu, Jiahong},
TITLE = {Global solutions of the 2{D} dissipative quasi-geostrophic equation in {B}esov spaces},
JOURNAL = {{\rm SIAM} J. Math. Anal.},
FJOURNAL = {SIAM Journal on Mathematical Analysis},
VOLUME = {36},
YEAR = {2004/05},
NUMBER = {3},
PAGES = {1014--1030},
ISSN = {0036-1410},
MRCLASS = {76B03 (35Q35 76U05 86A05)},
MRNUMBER = {2005i:76014},
MRREVIEWER = {Rapha{ë}l Danchin},
DOI = {10.1137/S0036141003435576},
ZBLNUMBER = {1083.76064},
} -
[Wu_2regularisations]
J. Wu, "The quasi-geostrophic equation and its two regularizations," Comm. Partial Differential Equations, vol. 27, iss. 5-6, pp. 1161-1181, 2002.
@article {Wu_2regularisations,
author = {Wu, Jiahong},
TITLE = {The quasi-geostrophic equation and its two regularizations},
JOURNAL = {Comm. Partial Differential Equations},
FJOURNAL = {Communications in Partial Differential Equations},
VOLUME = {27},
YEAR = {2002},
NUMBER = {5-6},
PAGES = {1161--1181},
ISSN = {0360-5302},
CODEN = {CPDIDZ},
MRCLASS = {76U05 (35K55 35Q35 76B99 86A05)},
MRNUMBER = {2003e:76114},
MRREVIEWER = {Jinqiao Duan},
DOI = {10.1081/PDE-120004898},
ZBLNUMBER = {1012.35067},
} -
[Wu_Holder]
J. Wu, "Solutions of the 2D quasi-geostrophic equation in Hölder spaces," Nonlinear Anal., vol. 62, iss. 4, pp. 579-594, 2005.
@article {Wu_Holder,
author = {Wu, Jiahong},
TITLE = {Solutions of the 2{D} quasi-geostrophic equation in {H}ölder spaces},
JOURNAL = {Nonlinear Anal.},
FJOURNAL = {Nonlinear Analysis. Theory, Methods \& Applications. An International Multidisciplinary Journal. Series A: Theory and Methods},
VOLUME = {62},
YEAR = {2005},
NUMBER = {4},
PAGES = {579--594},
ISSN = {0362-546X},
CODEN = {NOANDD},
MRCLASS = {76B03 (35Q35 76U05)},
MRNUMBER = {2006a:76011},
MRREVIEWER = {Isabelle Gallagher},
DOI = {10.1016/j.na.2005.03.053},
} -
[Schonbek^2]
M. E. Schonbek and T. P. Schonbek, "Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows," Discrete Contin. Dyn. Syst., vol. 13, iss. 5, pp. 1277-1304, 2005.
@article {Schonbek^2,
author = {Schonbek, M. E. and Schonbek, T. P.},
TITLE = {Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows},
JOURNAL = {Discrete Contin. Dyn. Syst.},
FJOURNAL = {Discrete and Continuous Dynamical Systems. Series A},
VOLUME = {13},
YEAR = {2005},
NUMBER = {5},
PAGES = {1277--1304},
ISSN = {1078-0947},
MRCLASS = {35Q35 (76B03)},
MRNUMBER = {2166670},
DOI = {10.3934/dcds.2005.13.1277},
ZBLNUMBER = {1091.35070},
} -
[Schonbek^1_2003]
M. E. Schonbek and T. P. Schonbek, "Asymptotic behavior to dissipative quasi-geostrophic flows," SIAM J. Math. Anal., vol. 35, iss. 2, pp. 357-375, 2003.
@article {Schonbek^1_2003,
author = {Schonbek, M. E. and Schonbek, T. P.},
TITLE = {Asymptotic behavior to dissipative quasi-geostrophic flows},
JOURNAL = {{\rm SIAM} J. Math. Anal.},
FJOURNAL = {SIAM Journal on Mathematical Analysis},
VOLUME = {35},
YEAR = {2003},
NUMBER = {2},
PAGES = {357--375},
ISSN = {0036-1410},
MRCLASS = {76U05 (35B40 35Q35 76B03 86A05)},
MRNUMBER = {2004e:76112},
MRREVIEWER = {Beno\^ ı t P. Desjardins},
DOI = {10.1137/S0036141002409362},
ZBLNUMBER = {1126.76386},
} -
[Resnick] S. Resnick, "Dynamical problems in nonlinear advective partial differential equations," PhD Thesis , University of Chicago, 1995.
@phdthesis {Resnick,
author = {Resnick, S.},
TITLE = {Dynamical problems in nonlinear advective partial differential equations},
TYPE = {Ph.D. thesis},
SCHOOL={University of Chicago},
YEAR={1995},
} -
[kislev] A. Kiselev, F. Nazarov, and A. Volberg, "Global well-posedness for the critical 2D dissipative quasi-geostrophic equation," Invent. Math., vol. 167, pp. 445-453, 2007.
@article{kislev,
author = {A. Kiselev and F. Nazarov and A. Volberg},
TITLE = {Global well-posedness for the critical 2{D} dissipative quasi-geostrophic equation},
JOURNAL = {Invent. Math.},
VOLUME={167},
YEAR={2007},
PAGES={445--453},
MRNUMBER={2008f:35408},
MRREVIEWER={Xiaoming Wang},
} -
[Vasseur] A. Vasseur, "A new proof of partial regularity of solutions to Navier-Stokes equations.," Nonlinear Differential Equations Appl., vol. 14, pp. 753-785, 2007.
@article {Vasseur,
author = {Vasseur, Alexis},
TITLE = {A new proof of partial regularity of solutions to {N}avier-{S}tokes equations.},
JOURNAL = {Nonlinear Differential Equations Appl.},
VOLUME={14},
YEAR={2007},
PAGES={753--785},
MRNUMBER={2009f:35257},
ZBLNUMBER = {1142.35066},
} -
[Mellet] A. Mellet and A. Vasseur, "$L^p$ estimates for quantities advected by a compressible flow," , preprint.
@techreport{Mellet,
author = {Mellet, Antoine and Vasseur, Alexis},
TITLE = {{$L^p$} estimates for quantities advected by a compressible flow},
TYPE = {preprint},
ZBLNUMBER = {05563814},
} -
[DeGiorgi] E. De Giorgi, "Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari," Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., vol. 3, pp. 25-43, 1957.
@article {DeGiorgi,
author = {De Giorgi, Ennio},
TITLE = {Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari},
JOURNAL = {Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat.},
VOLUME = {3},
YEAR = {1957},
PAGES = {25--43},
MRCLASS = {35.00 (49.00)},
MRNUMBER = {20 \#172},
MRREVIEWER = {C. B. Morrey, Jr.},
ZBLNUMBER = {0084.31901},
} -
[CS] L. Caffarelli and L. Silvestre, "An extension problem related to the fractional laplacian," Comm. Partial Differential Equations, vol. 32, pp. 1245-1260, 2007.
@article {CS,
author = {Caffarelli, Luis and Silvestre, Luis},
TITLE = {An extension problem related to the fractional laplacian},
JOURNAL = {Comm. Partial Differential Equations},
VOLUME={32},
YEAR={2007},
PAGES={1245--1260},
ZBLNUMBER = {1143.26002},
}