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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.

MSC:
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35B40 Asymptotic behavior of solutions to PDEs
35D30 Weak solutions to PDEs
35B45 A priori estimates in context of PDEs
86A05 Hydrology, hydrography, oceanography
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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[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
[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
[4] Chae D. On the regularity conditions for the dissipative quasi-geostrophic equations. SIAM J Math Anal, 2006, 37: 1649–1656 · Zbl 1141.76010
[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
[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
[8] Chen Z, Price W. Stability and instability analyses of the dissipative quasi-geostrophic equation. Nonlinearity, 2008, 21: 765–782 · Zbl 1133.76024
[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
[10] Constantin P, Wu J. Behavior of solutions of 2D quasi-geostrophic equations. SIAM J Math Anal, 1999, 30: 937–948 · Zbl 0957.76093
[11] Córdoba D. Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann of Math, 1998, 148: 1135–1152 · Zbl 0920.35109
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[30] Wu J. The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity, 2005, 18: 139–154 · Zbl 1067.35002
[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
[32] Zhou Y. Asymptotic stability for the 3D Navier-Stokes equations. Comm Partial Differential Equations, 2005, 30: 323–333 · Zbl 1142.35548
[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
[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
[35] Zhou Y. Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows. Nonlinearity, 2008, 21: 2061–2071 · Zbl 1186.35170
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