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Decay of weak solutions to the 2D dissipative quasi-geostrophic equation. (English) Zbl 1194.76040
Summary: We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data $$\theta_{0}$$ is in $$L^{2}$$ only, we prove that the $$L^{2}$$ norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For $$\theta_{0}$$ in $$L^{p} \cap L^{2}$$, with $$1 \leq p < 2$$, we are able to obtain a uniform decay rate in $$L^{2}$$. We also prove that when the $$L^{\frac{2}{2\alpha-1}}$$ norm of $$\theta_{0}$$ is small enough, the $$L^{q}$$ norms, for $$q > {\frac{2}{2\alpha-1}}$$, have uniform decay rates. This result allows us to prove decay for the $$L^{q}$$ norms, for $$q \geq {\frac{2}{2\alpha-1}}$$, when $$\theta_{0}$$ is in $$L^2 \cap L^{\frac{2}{2\alpha-1}}$$.

##### MSC:
 76D05 Navier-Stokes equations for incompressible viscous fluids 35B40 Asymptotic behavior of solutions to PDEs 35Q35 PDEs in connection with fluid mechanics 76U05 General theory of rotating fluids 86A05 Hydrology, hydrography, oceanography 86A10 Meteorology and atmospheric physics
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##### References:
 [1] Berselli L.C. (2002). Vanishing viscosity limit and long-time behaviour for 2D quasi-geostrophic equations. Indiana Univ. Math. J. 51: 905–930 · Zbl 1044.35055 [2] Cafarelli L., Kohn H. and Nirenberg L. (1982). Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35: 771–831 · Zbl 0509.35067 [3] Carpio A. (1996). Large-time behaviour in incompressible Navier-Stokes equations. SIAM J. Math. Anal 27: 449–475 · Zbl 0845.76019 [4] Carrillo J. and Ferreira L.C.F. (2007). Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equations. Monatshefte für Math 15: 111–142 · Zbl 1126.35048 [5] Carrillo J. and Ferreira L.C.F. (2006). Convergence towards self-similar asymptotic behaviour for the dissipative quasi-geostrophic equations. Banach Center Publ. 74: 95–115 · Zbl 1121.35109 [6] Carrillo, J., Ferreira, L.C.F.: Asymptotic behaviour for the subcritical dissipative quasi-geostrophic equations. Preprint UAB, 2006 · Zbl 1121.35109 [7] Chae D. (2003). The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity 16(2): 479–495 · Zbl 1029.35006 [8] Chae D. and Lee J. (2003). Global well-posedness in the super-critical dissipative quasi-geostrophic equations. Commun. Math. Phys. 233(2): 297–311 · Zbl 1019.86002 [9] Constantin, P., Cordoba, D., Wu, J.: On the critical dissipative quasi-geostrophic equation. Indiana Univ. Math. J. 50, Special Issue, 97–107 (2001) · Zbl 0989.86004 [10] Constantin P., Majda A. and Tabak E. (1994). Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7: 1495–1533 · Zbl 0809.35057 [11] Constantin P. and Wu J. (1999). Behaviour of solutions of 2D Quasi-geostrophic equations. SIAM J. Math. Anal. 30: 937–948 (electronic) · Zbl 0957.76093 [12] Córdoba A. and Córdoba D. (2004). A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249: 511–528 · Zbl 1309.76026 [13] Heywood, J.: Open problems in the theory of Navier-Stokes equations of viscous incompressible flow. The Navier-Stokes equations (Oberwolfach, 1988), Lecture Notes in Math. 1431, Berlin: Springer, 1990. pp. 1–22 [14] Ju N. (2004). Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the Sobolev space. Commun. Math. Phys. 251: 365–376 · Zbl 1106.35061 [15] Ju N. (2005). The maximum principle and the global attractor for the 2D dissipative quasi-geostrophic equation. Commun. Math. Phys. 255: 161–182 · Zbl 1088.37049 [16] Ju N. (2005). On the two dimensional quasi-geostrophic equations. Indiana Univ. Math. J. 54: 897–926 · Zbl 1185.35189 [17] Kato T. (1984). Strong L p solutions of the Navier-Stokes equation in $${\mathbb{R}}^m$$ , with applications to weak solutions. Math. Z. 187: 471–480 · Zbl 0545.35073 [18] Kato T. and Fujita H. (1962). On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova 32: 243–260 · Zbl 0114.05002 [19] Ogawa T., Rajopadhye S. and Schonbek M. (1997). Energy decay for a weak solution of the Navier-Stokes equation with slowly varying external forces. J. Funct. Anal. 144: 325–358 · Zbl 0873.35064 [20] Pedloskym J. (1987). Geophysical Fluid Dynamics. Springer Verlag, New York [21] Resnick, S.: Dynamical problems in non-linear advective partial differential equarions. Ph. D. Thesis, University of Chicago, 1995 [22] Schonbek M. (1985). L 2 decay for weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal. 88: 209–222 · Zbl 0602.76031 [23] Schonbek M. (1986). Large time behaviour of solutions to the Navier-Stokes equations. Comm. Partial Diff. Eqs. 11: 733–763 · Zbl 0607.35071 [24] Schonbek M. and Schonbek T. (2003). Asymptotic behavior to dissipative quasi-geostrophic flows. SIAM J. Math. Anal. 35: 357–375 (electronic) · Zbl 1126.76386 [25] Schonbek M. and Schonbek T. (2005). Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows. Disc. Contin. Dyn. Syst. 13: 1277–1304 · Zbl 1091.35070 [26] Serrin, J.: The initial value problem for the Navier-Stokes equations. In: Nonlinear problems (Proc. Sympos., Madison, Wis.), Madison, WI: Oniv. ofvisc. Press, pp. 69–98, 1963 · Zbl 0115.08502 [27] Wu, J.: Dissipative quasi-geostrophic equations with L p data. Electron. J. Diff. Eq. (2001), No. 56, 13 pp. (electronic) · Zbl 0987.35127 [28] Wu J. (2002). The 2D dissipative quasi-geostrophic equation. Appl. Math. Letters 15: 925–930 · Zbl 1016.35060 [29] Wu J. (2002). The quasi-geostrophic equation and its two regularizations. Comm. Partial Diff. Eqs. 27(5-6): 1161–1181 · Zbl 1012.35067 [30] Wu J. (2004). Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces. SIAM J. Math. Anal. 36(3): 1014–1030. (electronic) · Zbl 1083.76064 [31] Wu J. (2005). Solutions of the 2D quasi-geostrophic equation in Hölder spaces. Nonlinear Anal. 62(4): 579–594 · Zbl 1116.35348 [32] Wu J. (2005). The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation. Nonlinearity 18(1): 139–154 · Zbl 1067.35002 [33] Zhang L. (1995). Sharp rates of decay of solutions to 2-dimensional Navier-Stokes equations. Comm. Partial Diff. Eqs. 20: 119–127 · Zbl 0823.35145
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