zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Asymptotic profile of solutions to the two-dimensional dissipative quasi-geostrophic equation. (English) Zbl 1214.35049
Summary: This paper is concerned with the asymptotic behavior of the two-dimensional dissipative quasi-geostrophic equation. Based on the spectral decomposition of the Laplacian operator and iterative techniques, we obtain improved $L^2$ decay rates of weak solutions and derive more explicit upper bounds of higher order derivatives of solutions. We also prove the asymptotic stability of the subcritical quasi-geostrophic equation under large initial and external perturbations.

35Q35PDEs in connection with fluid mechanics
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
35B40Asymptotic behavior of solutions of PDE
35D30Weak solutions of PDE
35B45A priori estimates for solutions of PDE
86A05Hydrology, hydrography, oceanography
35B30Dependence of solutions of PDE on initial and boundary data, parameters
Full Text: DOI
[1] Caffarelli L, Kohn R, Nirenberg L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun Pure Appl Math, 1982, 35: 771--831 · Zbl 0509.35067 · doi:10.1002/cpa.3160350604
[2] Caffarelli L, Vasseur A. Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann of Math, in press · Zbl 1204.35063
[3] Carrillo J, Lucas L. The asymptotic behaviour of subcritical dissipative quasi-geostrophic equations. Nonlinearity, 2008, 21: 1001--1018 · Zbl 1136.76052 · doi:10.1088/0951-7715/21/5/006
[4] Chae D. On the regularity conditions for the dissipative quasi-geostrophic equations. SIAM J Math Anal, 2006, 37: 1649--1656 · Zbl 1141.76010 · doi:10.1137/040616954
[5] Chae D, Lee J. Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Commun Math Phys, 2003, 233: 297--311 · Zbl 1019.86002
[6] Chen Q, Miao C, Zhang Z. A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation. Commun Math Phys, 2007, 271: 821--838 · Zbl 1142.35069 · doi:10.1007/s00220-007-0193-7
[7] Chen Z, Ghil M, Siminnet E, et al. Hopf bifurcation in quasi-geostrophic flow. SIAM J Appl Math, 2003, 64: 343--368 · Zbl 1126.76327 · doi:10.1137/S0036139902406164
[8] Chen Z, Price W. Stability and instability analyses of the dissipative quasi-geostrophic equation. Nonlinearity, 2008, 21: 765--782 · Zbl 1133.76024 · doi:10.1088/0951-7715/21/4/006
[9] Constantin P, Majda A, Tabak E. Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity, 1994, 7: 1495--1533 · Zbl 0809.35057 · doi:10.1088/0951-7715/7/6/001
[10] Constantin P, Wu J. Behavior of solutions of 2D quasi-geostrophic equations. SIAM J Math Anal, 1999, 30: 937--948 · Zbl 0957.76093 · doi:10.1137/S0036141098337333
[11] Córdoba D. Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann of Math, 1998, 148: 1135--1152 · Zbl 0920.35109 · doi:10.2307/121037
[12] Córdoba A, Córdoba D. A maximum principle applied to quasi-geostrophic equations. Commun Math Phys, 2004, 249: 511--528 · Zbl 1309.76026
[13] Dong B, Chen Z. Asymptotic stability of non-Newtonian flows with large perturbation. Appl Math Comput, 2006, 173: 243--250 · Zbl 1138.35380 · doi:10.1016/j.amc.2005.04.002
[14] Dong B, Chen Z. Remarks on upper and lower bounds of solutions to the Navier-Stokes equations in $\mathbb{R}$2. Appl Math Comput, 2006, 182: 553--558 · Zbl 1103.76016 · doi:10.1016/j.amc.2006.04.017
[15] Dong B, Chen Z. Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation. Nonlinearity, 2006, 19: 2919--2928 · Zbl 1109.76063 · doi:10.1088/0951-7715/19/12/011
[16] Dong B, Chen Z. A remark on regularity criterion for the dissipative quasi geostrophic equations. J Math Anal Appl, 2007, 329: 1212--1217 · Zbl 1154.76339 · doi:10.1016/j.jmaa.2006.07.054
[17] Dong B, Jiang W. On the decay of higher order derivatives of solutions to Ladyzhenskaya model for incompressible viscous flows. Sci China Ser A, 2008, 51: 925--934 · Zbl 1153.35062 · doi:10.1007/s11425-007-0196-z
[18] Ju N. Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equation in the Sobolev space. Commun Math Phys, 2004, 251: 365--376 · Zbl 1106.35061 · doi:10.1007/s00220-004-1062-2
[19] Ju N. The maximum principle and the global attractor for the 2D dissipative quasi-geostrophic equation. Commun Math Phys, 2005, 255: 161--182 · Zbl 1088.37049 · doi:10.1007/s00220-004-1256-7
[20] Kajikiya R, Miyakawa T. On L 2 decay of weak solutions of the Navier-Stokes Equations in $\mathbb{R}$n. Math Z, 1986, 192: 135--148 · Zbl 0607.35072 · doi:10.1007/BF01162027
[21] Kiselev A, Nazarov F, Volberg A. Global well-posedness for the critical 2D dissipative quasi-geostrophic equation. Invent Math, 2007, 167: 445--453 · Zbl 1121.35115 · doi:10.1007/s00222-006-0020-3
[22] Kozono H. Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations. J Funct Anal, 2000, 176: 153--197 · Zbl 0970.35106 · doi:10.1006/jfan.2000.3625
[23] Majda A, Tabak E. A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Phys D, 1996, 98: 515--522 · Zbl 0899.76105 · doi:10.1016/0167-2789(96)00114-5
[24] Niche C, Schonbek M. Decay of weak solutions to the 2D dissipative quasi-geostrophic equation. Commun Math Phys, 2007, 276: 93--115 · Zbl 1194.76040 · doi:10.1007/s00220-007-0327-y
[25] Oliver M, Titi E S. Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in $\mathbb{R}$n. J Funct Anal, 2000, 172: 1--18 · Zbl 0960.35081 · doi:10.1006/jfan.1999.3550
[26] Pedlosky J. Geophysical Fluid Dynamics. New York: Springer, 1987 · Zbl 0713.76005
[27] Resnick S. Dynamical problems in non-linear advective partial differential equarions. PhD Thesis. Chicago: University of Chicago, 1995
[28] Schonbek M, Schonbek T. Asymptotic behavior to dissipative quasi-geostrophic flows. SIAM J Math Anal, 2003, 35: 357--375 · Zbl 1126.76386 · doi:10.1137/S0036141002409362
[29] Schonbek M, Schonbek T. Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Discrete Contin Dyn Syst, 2005, 13: 1277--1304 · Zbl 1091.35070 · doi:10.3934/dcds.2005.13.1277
[30] Wu J. The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity, 2005, 18: 139--154 · Zbl 1067.35002 · doi:10.1088/0951-7715/18/1/008
[31] Zhou Y. On the energy and helicity conservations for the 2D quasi-geostrophic equation. Ann Henri Poincaré, 2005, 6: 791--799 · Zbl 1077.76010 · doi:10.1007/s00023-005-0223-y
[32] Zhou Y. Asymptotic stability for the 3D Navier-Stokes equations. Comm Partial Differential Equations, 2005, 30: 323--333 · Zbl 1142.35548 · doi:10.1081/PDE-200037770
[33] Zhou Y. Decay rate of higher order derivatives for solutions to the 2D dissipative quasi-geostrophic flows. Discrete Contin Dyn Syst, 2006, 14: 525--532 · Zbl 1185.35203 · doi:10.3934/dcds.2006.14.525
[34] Zhou Y. A remark on the decay of solutions to the 3-D Navier-Stokes equations. Math Methods Appl Sci, 2007, 30: 1223--1229 · Zbl 1117.35064 · doi:10.1002/mma.841
[35] Zhou Y. Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows. Nonlinearity, 2008, 21: 2061--2071 · Zbl 1186.35170 · doi:10.1088/0951-7715/21/9/008