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Solutions of the 2D quasi-geostrophic equation in Hölder spaces. (English) Zbl 1116.35348
Summary: The 2D quasi-geostrophic equation \[ \partial_t\theta+u\cdot\nabla \theta+\kappa(-\Delta)^\alpha\theta=0,\quad u={\mathcal R}^\perp (\theta) \] is a two-dimensional model of the 3D hydrodynamics equations. When \(\alpha\leq\frac 12\), the issue of existence and uniqueness concerning this equation becomes difficult. It is shown here that this equation with either \(\square=0\) or \(\square>0\) and \(0\leq \alpha \leq\frac 12\) has a unique local in time solution corresponding to any initial datum in the space \(C^rL^q\) for \(r<1\) and \(\varphi>1\).

MSC:
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76U05 General theory of rotating fluids
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[1] Berselli, L., Vanishing viscosity limit and long-time behavior for 2D quasi-geostrophic equations, Indiana univ. math. J., 51, 905-930, (2002) · Zbl 1044.35055
[2] Chae, D., The quasi-geostrophic equation in the triebel – lizorkin spaces, Nonlinearity, 16, 479-495, (2003) · Zbl 1029.35006
[3] 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
[4] Chemin, J.-Y., Perfect incompressible fluids, (1998), Clarendon Press Oxford
[5] Constantin, P.; Córdoba, D.; Wu, J., On the critical dissipative quasi-geostrophic equation, Indiana univ. math. J., 50, 97-107, (2001) · Zbl 0989.86004
[6] 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
[7] Constantin, P.; Wu, J., Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. math. anal., 30, 937-948, (1999) · Zbl 0957.76093
[8] Córdoba, D., Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. math., 148, 1135-1152, (1998) · Zbl 0920.35109
[9] A. Córdoba, D. Córdoba, A maximum principle applied to quasi-geostrophic equations, preprint.
[10] Córdoba, D.; Fefferman, C., Growth of solutions for QG and 2D Euler equations, J. amer. math. soc., 15, 665-670, (2002) · Zbl 1013.76011
[11] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer New York · Zbl 0713.76005
[12] S. Resnick, Dynamical problem in nonlinear advective partial differential equations, Ph.D. Thesis, University of Chicago, Chicago, 1995.
[13] M. Schonbek, T. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal. 35 (2003) 357-375. · Zbl 1126.76386
[14] Wu, J., Inviscid limits and regularity estimates for the solutions of the 2D dissipative quasi-geostrophic equations, Indiana univ. math. J., 46, 1113-1124, (1997) · Zbl 0909.35111
[15] Wu, J., Dissipative quasi-geostrophic equations with \(L^p\) data, Electron. J. differential equations, 2001, 1-13, (2001) · Zbl 0987.35127
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