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Stability analysis of the supercritical surface quasi-geostrophic equation. (English) Zbl 1470.35285

Summary: This paper is devoted to the study of the stability issue of the supercritical dissipative surface quasi-geostrophic equation with nondecay low-regular external force. Supposing that the weak solution \(\theta(x, t)\) of the surface quasi-geostrophic equation with the force \(f \in L^2(0, T; H^{- \alpha / 2}(\mathbb{R}^2))\) satisfies the growth condition in the critical BMO space \(\nabla \theta \in L^1(0, \infty; \mathrm{BMO})\), it is proved that every perturbed weak solution \(\overline{\theta}(t)\) converges asymptotically to solution \(\theta(t)\) of the original surface quasi-geostrophic equation. The initial and external forcing perturbations are allowed to be large.

MSC:

35Q35 PDEs in connection with fluid mechanics
86A05 Hydrology, hydrography, oceanography
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[1] Pedlosky, J., Geophysical Fluid Dynamics (1987), New York, NY, USA: Springer, New York, NY, USA · Zbl 0713.76005
[2] Constantin, P.; Majda, A. J.; Tabak, E., Formation of strong fronts in the \(n\)-D quasigeostrophic thermal active scalar, Nonlinearity, 7, 6, 1495-1533 (1994) · Zbl 0809.35057
[3] Majda, A. J.; Tabak, E. G., A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow, Physica D, 98, 2-4, 515-522 (1996) · Zbl 0899.76105
[4] Temam, R., Navier-Stokes Equations. Theory and Numerical Analysis, x+500 (1977), Amsterdam, The Netherlands: North-Holland Publishing, Amsterdam, The Netherlands
[5] Constantin, P.; Wu, J., Behavior of solutions of 2D quasi-geostrophic equations, SIAM Journal on Mathematical Analysis, 30, 5, 937-948 (1999) · Zbl 0957.76093
[6] Kiselev, A.; Nazarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Inventiones Mathematicae, 167, 3, 445-453 (2007) · Zbl 1121.35115
[7] Caffarelli, L. A.; Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Mathematics, 171, 3, 1903-1930 (2010) · Zbl 1204.35063
[8] Castro, A.; Córdoba, D., Infinite energy solutions of the surface quasi-geostrophic equation, Advances in Mathematics, 225, 4, 1820-1829 (2010) · Zbl 1205.35219
[9] Chen, Q.; Miao, C.; Zhang, Z., A new Bernstein’s inequality and the 2D dissipative quasi-geostrophic equation, Communications in Mathematical Physics, 271, 3, 821-838 (2007) · Zbl 1142.35069
[10] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Communications in Mathematical Physics, 249, 3, 511-528 (2004) · Zbl 1309.76026
[11] Constantin, P.; Wu, J., Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations, Annales de l’Institut Henri Poincaré (C), 26, 1, 159-180 (2009) · Zbl 1163.76010
[12] Dong, B.-Q.; Chen, Z.-M., On the weak-strong uniqueness of the dissipative surface quasi-geostrophic equation, Nonlinearity, 25, 5, 1513-1524 (2012) · Zbl 1241.35158
[13] Dong, H.; Pavlović, N., Regularity criteria for the dissipative quasi-geostrophic equations in Hölder spaces, Communications in Mathematical Physics, 290, 3, 801-812 (2009) · Zbl 1185.35187
[14] Ju, N., Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space, Communications in Mathematical Physics, 251, 2, 365-376 (2004) · Zbl 1106.35061
[15] Wang, H.; Zhang, Z., A frequency localized maximum principle applied to the 2D quasi-geostrophic equation, Communications in Mathematical Physics, 301, 1, 105-129 (2011) · Zbl 1248.35211
[16] Wu, J., Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM Journal on Mathematical Analysis, 36, 3, 1014-1030 (2005) · Zbl 1083.76064
[17] Chae, D.; Constantin, P.; Wu, J., Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations, Archive for Rational Mechanics and Analysis, 202, 1, 35-62 (2011) · Zbl 1266.76010
[18] Chae, D.; Constantin, P.; Córdoba, D.; Gancedo, F.; Wu, J., Generalized surface quasi-geostrophic equations with singular velocities, Communications on Pure and Applied Mathematics, 65, 8, 1037-1066 (2012) · Zbl 1244.35108
[19] Fefferman, C.; Rodrigo, J. L., Analytic sharp fronts for the surface quasi-geostrophic equation, Communications in Mathematical Physics, 303, 1, 261-288 (2011) · Zbl 1228.35010
[20] Khouider, B.; Titi, E. S., An inviscid regularization for the surface quasi-geostrophic equation, Communications on Pure and Applied Mathematics, 61, 10, 1331-1346 (2008) · Zbl 1149.35018
[21] Miao, C.; Xue, L., Global well-posedness for a modified critical dissipative quasi-geostrophic equation, Journal of Differential Equations, 252, 1, 792-818 (2012) · Zbl 1382.35233
[22] Dong, B.-Q.; Liu, Q., Asymptotic profile of solutions to the two-dimensional dissipative quasi-geostrophic equation, Science China. Mathematics, 53, 10, 2733-2748 (2010) · Zbl 1214.35049
[23] Niche, C. J.; Schonbek, M. E., Decay of weak solutions to the 2D dissipative quasi-geostrophic equation, Communications in Mathematical Physics, 276, 1, 93-115 (2007) · Zbl 1194.76040
[24] Zhou, Y., Decay rate of higher order derivatives for solutions to the 2-D dissipative quasi-geostrophic flows, Discrete and Continuous Dynamical Systems, 14, 3, 525-532 (2006) · Zbl 1185.35203
[25] Zhou, Y., Asymptotic behaviour of the solutions to the 2D dissipative quasi-geostrophic flows, Nonlinearity, 21, 9, 2061-2071 (2008) · Zbl 1186.35170
[26] Chae, D.; Lee, J., Global well-posedness in the super-critical dissipative quasi-geostrophic equations, Communications in Mathematical Physics, 233, 2, 297-311 (2003) · Zbl 1019.86002
[27] Chen, Z.-M.; Price, W. G., Stability and instability analyses of the dissipative quasi-geostrophic equation, Nonlinearity, 21, 4, 765-782 (2008) · Zbl 1133.76024
[28] Dong, B.-Q.; Chen, Z.-M., Asymptotic stability of the critical and super-critical dissipative quasi-geostrophic equation, Nonlinearity, 19, 12, 2919-2928 (2006) · Zbl 1109.76063
[29] Dong, B.-Q.; Chen, Z.-M., A remark on regularity criterion for the dissipative quasi-geostrophic equations, Journal of Mathematical Analysis and Applications, 329, 2, 1212-1217 (2007) · Zbl 1154.76339
[30] Kozono, H., Asymptotic stability of large solutions with large perturbation to the Navier-Stokes equations, Journal of Functional Analysis, 176, 2, 153-197 (2000) · Zbl 0970.35106
[31] Zhang, L., New results of general \(n\)-dimensional incompressible Navier-Stokes equations, Journal of Differential Equations, 245, 11, 3470-3502 (2008) · Zbl 1157.35084
[32] Zhou, Y., Asymptotic stability for the 3D Navier-Stokes equations, Communications in Partial Differential Equations, 30, 1-3, 323-333 (2005) · Zbl 1142.35548
[33] Marchand, F., Weak-strong uniqueness criteria for the critical quasi-geostrophic equation, Physica D, 237, 10-12, 1346-1351 (2008) · Zbl 1143.76339
[34] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations. Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, viii+279 (1983), New York, NY, USA: Springer, New York, NY, USA · Zbl 0516.47023
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