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Global solutions to the two-dimensional quasi-geostrophic equation with critical or super-critical dissipation. (English) Zbl 1145.76053
Summary: The two-dimensional quasi-geostrophic equation with critical and super-critical dissipation is studied in Sobolev space \(H^{s}(\mathbb R^{2})\). For critical case (\(\alpha = \frac12\)), existence of global (large) solutions in \(H^{s}\) is proved for \(s \geq \frac12\) when \(\|\Theta_0 \|_{L^\infty}\) is small. This generalizes and improves the results of P. Constantin, D. Cordoba and J. Wu [Indiana Univ. Math. J. 50, Spec. Iss., 97–107 (2001; Zbl 0989.86004)] for \(s=1,2\) and the result of A. Cordoba and D. Cordoba [Commun. Math. Phys. 249, No. 3, 511–528 (2004; Zbl 1309.76026)] for \(s = \frac32\). For \(s \geq 1\), these solutions are also unique. The improvement for pushing \(s\) down from 1 to \(\frac12\) is somewhat surprising and unexpected. For super-critical case (\(\alpha \in (0,\frac12)\)), existence and uniqueness of global (large) solution in \(H^{s}\) is proved when the product \(\|\Theta_0 \|_{H^s}^\beta \|\Theta_0 \|_{L^p}^{1-\beta}\) is small for suitable \(s \geq 2-2\alpha\), \(p \in [1,\infty]\) and \(\beta \in (0,1]\).

76U05 General theory of rotating fluids
35Q35 PDEs in connection with fluid mechanics
86A05 Hydrology, hydrography, oceanography
Full Text: DOI
[1] Chae, D.: The Quasi-Geostrophic Equation in the Triebel-Lizorkin Spaces. Nonlinearity, 16, 479–495 (2003) · Zbl 1029.35006
[2] Chae, D., Lee, J.: Global well-posedness in the super-critical dissipative quasi-geostrophic Equations. Comm. Math. Phys. 233(2), 297–311 (2003) · Zbl 1019.86002
[3] Coifman, R., Meyer, Y.: Au delĂ  des operateurs pseudo-differentiels. Asterisque 57, Societe Mathematique de France, Paris, 1978
[4] Constantin, P., Cordoba, D., Wu, J.: On the critical dissipative quasi-geostrophic equations. Indiana University Mathematics Journal, 50, 97–107 (2001) · Zbl 0989.86004
[5] 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
[6] Constantin, P., Wu, J.: Behavior of solutions of 2D quasi-geostrophic equations. SIAM Journal on Mathematical Analysis, 30, 937–948 (1999) · Zbl 0957.76093
[7] Cordoba, D.: Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. of Math. 148, 1135–1152 (1998) · Zbl 0920.35109
[8] Cordoba, A., Cordoba, D.: A maximum principle applied to quasi-geostrophic equations. Comm. Math. Phys. 249(3), 511–528 (2004) · Zbl 0438.42013
[9] Cordoba, D., Fefferman, C.: Growth of solutions for QG and 2D Euler equations. J. Amer. Math. Soc. 15(3), 665–670 (2002) · Zbl 1013.76011
[10] Deng J., Hou, Y., Yu, X.: Geometric Properties and Non-blowup of 2-D Quasi-geostrophic Equation. 2004, preprint
[11] Friedlander, S.: On vortex tube stretching and instabilities in an inviscid fluid. J. Math. Fluid Mech. 4(1), 30–44 (2002) · Zbl 0992.35078
[12] Held, I., Pierrehumbert, R., Garner, S., Swanson, K.: Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20 (1995) · Zbl 0832.76012
[13] Ju, N.: Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equations in the sobolev space. Comm. Math. Phys. 251(2), 365–376 (2004) · Zbl 1106.35061
[14] Ju, N.: The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations. Comm. Math. Phys. 255(1), 161–181 (2005) · Zbl 1088.37049
[15] Ju, N.: On the two dimensional Quasi-Geostrophic equation. Indiana Univ. Math. J. 54(3), 897–926 (2005) · Zbl 1185.35189
[16] Ju, N.: Geometric constrains for global regularity of 2D quasi-geostrophic flow. preprint, submitted · Zbl 1098.35046
[17] Ju, N.: Dissipative quasi-geostrophic equation: local well-posedness, global regularity and similarity solutions. preprint submitted · Zbl 1129.35062
[18] Kato, T.: Liapunov functions and monotonicity in the Navier-Stokes equations. Lecture Notes in Mathematics, 1450, Springer-Verlag, Berlin, 1990 · Zbl 0727.35107
[19] Kato, T., Ponce, G.: Commutator estimates and Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988) · Zbl 0671.35066
[20] Kenig, C., Ponce, G., Vega, L.: Well-posedness of the initial value problem for the Korteweg-De Vries equation. J. Amer. Math. Soc. 4, 323–347 (1991) · Zbl 0737.35102
[21] Majda, A., Tabak, E.: A two-dimensional model for quasi-geostrophic flow: comparison with the two-dimensional Euler flow. Nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995). Phys. D 98(2–4), 515–522 (1996) · Zbl 0899.76105
[22] Ohkitani, K., Yamada, M.: Inviscid and inviscid limit behavior of a surface quasi-geostrophic flow. Phys. Fluids, 9, 876–882 (1997) · Zbl 1185.76841
[23] Pedlosky, J.: Geophysical fluid dynamics. Springer-Verlag, New York, 1987 · Zbl 0713.76005
[24] Resnick, S.: Dynamical problems in non-linear advective partial differential equations. Ph.D. thesis, University of Chicago, 1995
[25] Schonbek, M., Schonbek, T.: Asymptotic behavior to dissipative quasi-geostrophic flows. SIAM J. Math. Anal. 35, 357–375 (2003) · Zbl 1126.76386
[26] Stein, E.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, NJ, 1970 · Zbl 0207.13501
[27] Wu, J.: The Quasi-Geostrophic Equation and Its Two regularizations, communications in Partial Differential Equations. 27, 1161–1181 (2002) · Zbl 1012.35067
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