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Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation. (English) Zbl 1122.76014
Summary: This paper examines Besov space solutions of the 2-D quasi-geostrophic (QG) equation with dissipation induced by a fractional Laplacian \((-\Delta)^\alpha\). The goal is threefold: first, to extend a previous result on solutions in the inhomogeneous Besov space \(B_{2,q}^r\) [the author, SIAM J. Math. Anal. 36, 1014–1030 (2004–2005)] to cover the case when \(r=2-2\alpha\); second, to establish the global existence of solutions in homogeneous Besov space \(\mathring B_{p,q}^r\) with general indices \(p\) and \(q\); and third, to determine the uniqueness of solutions in any one of the four spaces: \(B_{2,q}^s,\mathring B_{p,q}^r, L^q((0,T); B_{2,q}^{s+\frac{2\alpha}{q}})\) and \(L^q((0,T);\mathring B_{p,q}^{r+\frac{2\alpha}{q}})\), where \(s\geq 2-2\alpha\) and \(r=1-2\alpha+ \frac 2p\).

MSC:
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
Keywords:
Besov space
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[1] Caffarelli, L.; Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, (2006)
[2] Chae, D., On the regularity conditions for the dissipative quasi-geostrophic equations, SIAM J. math. anal., 37, 1649-1656, (2006) · Zbl 1141.76010
[3] Chae, D.; Lee, J., Global well-posedness in the super-critical dissipative quasi-geostrophic equations, Comm. math. phys., 233, 297-311, (2003) · Zbl 1019.86002
[4] Chemin, J.-Y., Théorèmes d’unicité pour le système de navier – stokes tridimensionnel, J. anal. math., 77, 27-50, (1999) · Zbl 0938.35125
[5] Q. Chen, C. Miao, Z. Zhang, A new Bernstein inequality and the 2D dissipative quasi-geostrophic equation (preprint)
[6] Constantin, P., Euler equations, navier – stokes equations and turbulence, () · Zbl 1002.76038
[7] Constantin, P.; Córdoba, D.; Wu, J., On the critical dissipative quasi-geostrophic equation, Indiana univ. math. J., 50, 97-107, (2001) · Zbl 0989.86004
[8] Constantin, P.; Majda, A.; Tabak, E., Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity, 7, 1495-1533, (1994) · Zbl 0809.35057
[9] Constantin, P.; Wu, J., Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. math. anal., 30, 937-948, (1999) · Zbl 0957.76093
[10] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Comm. math. phys., 249, 511-528, (2004) · Zbl 1309.76026
[11] Ju, N., The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equations, Comm. math. phys., 255, 161-181, (2005) · Zbl 1088.37049
[12] Kiselev, A.; Nazarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, (2006) · Zbl 1121.35115
[13] Pedlosky, J., Geophysical fluid dynamics, (1987), Springer-Verlag New York · Zbl 0713.76005
[14] S. Resnick, Dynamical problems in nonlinear partial differential equations, Ph.D. Thesis, University of Chicago, 1995
[15] Schonbek, M.; Schonbek, T., Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. math. anal., 35, 357-375, (2003) · Zbl 1126.76386
[16] Schonbek, M.; Schonbek, T., Moments and lower bounds in the far-field of solutions to quasi-geostrophic flows, Discrete contin. dyn. syst., 13, 1277-1304, (2005) · Zbl 1091.35070
[17] Wu, J., Global solutions of the 2D dissipative quasi-geostrophic equation in Besov spaces, SIAM J. math. anal., 36, 1014-1030, (2004-2005)
[18] Wu, J., Lower bounds for an integral involving fractional Laplacians and the generalized navier – stokes equations in Besov spaces, Comm. math. phys., 263, 803-831, (2006) · Zbl 1104.35037
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