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A maximum principle applied to quasi-geostrophic equations. (English) Zbl 1309.76026
Summary: We study the initial value problem for dissipative 2D quasi-geostrophic equations proving local existence, global results for small initial data in the super-critical case, decay of $$L^p$$ -norms and asymptotic behavior of viscosity solution in the critical case. Our proofs are based on a maximum principle valid for more general flows.

##### MSC:
 76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids 35Q35 PDEs in connection with fluid mechanics 76U05 General theory of rotating fluids 86A05 Hydrology, hydrography, oceanography 26A33 Fractional derivatives and integrals
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##### References:
 [1] Baroud, Ch. N., Plapp, B.B., She, Z.-S., Swinney, H.L.: Anomalous self-similarity in a turbulent rapidly rotating fluid. Phys. Rev. Lett. 88, 114501 (2002) [2] Berselli, L.: Vanishing viscosity limit and long-time behavior for 2D Quasi-geostrophic equations. Indiana Univ. Math. J. 51 (4), 905–930 (2002) · Zbl 1044.35055 [3] Chae, D.: The quasi-geostrophic equation in the Triebel-Lizorkin spaces. Nonlinearity 16 (2), 479–495 (2003) · Zbl 1029.35006 [4] Chae, D., Lee, J.: Global Well-Posedness in the super critical dissipative Quasi-geostrophic equations. Commun. Math. Phys. 233, 297–311 (2003) · Zbl 1019.86002 [5] Coifman, R., Meyer, Y.: Au delà des operateurs pseudo-differentiels. Asterisqué 57, Paris: Société Mathmatique de France, 1978, pp. 154 [6] Coifman, R., Meyer, Y.: Ondelettes et operateurs. III. (French) [Wavelets and operators. III] Operateurs multilinaires. [Multilinear operators] Actualits Mathmatiques. [Current Mathematical Topics] Paris: Hermann, 1991 [7] Constantin, P.: Energy Spectrum of Quasi-geostrophic Turbulence. Phys. Rev. Lett. 89 (18), 1804501–4 (2002) [8] Constantin, P., Cordoba, D., Wu, J.: On the critical dissipative Quasi-geostrophic equation. Indiana Univ. Math. J. 50, 97–107 (2001) · Zbl 0989.86004 [9] Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994) · Zbl 0809.35057 [10] Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM J. Math. Anal. 30, 937–948 (1999) · Zbl 0957.76093 [11] Cordoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148, 1135–1152 (1998) · Zbl 0920.35109 [12] Cordoba, A., Cordoba, D.: A pointwise estimate for fractionary derivatives with applications to P.D.E. Proc. Natl. Acad. Sci. USA 100 (26), 15316–15317 (2003) · Zbl 1111.26010 [13] Cordoba, D., Fefferman, C.: Growth of solutions for QG and 2D Euler equations. J. Am. Math. Soc. 15 (3), 665–670 (2002) · Zbl 1013.76011 [14] Dinaburg, E.I., Posvyanskii, V.S., Sinai, Ya.G.: On some approximations of the Quasi-geostrophic equation. Preprint. · Zbl 1054.35050 [15] Held, I., Pierrehumbert, R., Garner, S., Swanson, K.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995) · Zbl 0832.76012 [16] Kato, T., Ponce, G.: Commutators estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988) · Zbl 0671.35066 [17] Pedlosky, J.: Geophysical Fluid Dynamics. New York: Springer-Verlag, 1987 · Zbl 0713.76005 [18] Resnick, S.: Dynamical problems in nonlinear advective partial differential equations. Ph.D. thesis, University of Chicago, Chicago 1995 [19] Schonbek, M.E., Schonbek, T.P.: Asymptotic behavior to dissipative quasi-geostrophic flows. SIAM J. Math. Anal. 35 (2), 357–375 (2003) · Zbl 1126.76386 [20] Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton NJ: Princeton University Press, 1970 · Zbl 0207.13501 [21] Stein, E., Zygmund, A.: Boundedness of translation invariant operators on Holder and Lp-spaces. Ann. of Math. 85, 337–349 (1967) · Zbl 0172.40102 [22] Wu, J.: Dissipative quasi-geostrophic equations with Lp data. Electronic J. Differ. Eq. 56, 1–13 (2001) · Zbl 0987.35127 [23] Wu, J.: The quasi-geostrophic equations and its two regularizations. Comm. Partial Differ. Eq. 27 (5–6), 1161–1181 (2002) · Zbl 1012.35067 [24] Wu, J.: Inviscid limits and regularity estimates for the solutions of the 2-D dissipative Quasi- geostrophic equations. Indiana Univ. Math. J. 46 (4), 1113–1124 (1997) · Zbl 0909.35111
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