zbMATH — the first resource for mathematics

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 \(r=1-2\alpha+ \frac 2p\).

76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
Besov space
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.