×

Commutator estimate in terms of partial derivatives of solutions for the dissipative quasi-geostrophic equation. (English) Zbl 1346.35028

Summary: Through Littlewood-Paley decomposition argument, a commutator estimate in terms of partial derivatives of solutions for the critical and supercritical dissipative 2D Quasi-Geostrophic equation is established. As an application of this estimate, we obtain some new a priori estimates and prove the existence and uniqueness of solution for the small initial data in critical Besov spaces.

MSC:

35B45 A priori estimates in context of PDEs
35G25 Initial value problems for nonlinear higher-order PDEs
86A05 Hydrology, hydrography, oceanography
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bae, H., Global well-posedness of dissipative quasi-geostrophic equations in the critical spaces, Proc. Amer. Math. Soc., 136, 257-261 (2008) · Zbl 1175.35108
[2] Caffarelli, L.; Vasseur, A., Drift diffusion equations with fractional diffusion quasi-geostrophic equations, Ann. of Math., 171, 1903-1930 (2006) · Zbl 1204.35063
[3] Cannone, M., A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoam., 13, 515-541 (1997) · Zbl 0897.35061
[4] Cannone, M.; Meyer, Y., Littlewood-Paley decomposition and Navier-Stokes equations, Methods Appl. Anal., 2, 307-319 (1995) · Zbl 0842.35074
[5] 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
[6] Chemin, J.-Y., Perfect Incompressible Fluids (1998), Oxford University Press: Oxford University Press New York
[7] Chen, Q.; Miao, C.; Zhang, Z., A new Bernsten’s inequality and the 2D dissipative quasi-geostrophic equation, Comm. Math. Phys., 271, 821-838 (2007) · Zbl 1142.35069
[8] Chen, Q.; Miao, C.; Zhang, Z., Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity, Comm. Pure Appl. Math., 63, 1173-1224 (2010) · Zbl 1202.35002
[9] Chen, Q.; Miao, C.; Zhang, Z., Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Rev. Mat. Iberoam., 26, 915-946 (2010) · Zbl 1205.35189
[10] Constantin, P.; Majda, A.; Tabak, E., Formation of strong fronts in the 2D quasi-geostrophic thermal active scalar, Nonlinearity, 7, 1495-1533 (1994)
[11] Constantin, P.; Wu, J., Behavior of solutions of 2D quasi-geostrophic equations, SIAM J. Math. Anal., 30, 937-948 (1999) · Zbl 0957.76093
[12] Constantin, P.; Córdoba, D.; Wu, J., On the critical dissipative quasi-geostrophic equation, Indiana Univ. Math. J., 50, 97-107 (2001) · Zbl 0989.86004
[13] Córdoba, A.; Córdoba, D., A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249, 511-528 (2004) · Zbl 1309.76026
[14] Danchin, R., Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh, 133A, 1311-1334 (2003) · Zbl 1050.76013
[15] Dong, H.; Li, D., On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces, J. Differential Equations, 248, 2684-2702 (2010) · Zbl 1193.35151
[16] Fefferman, C. L.; Mccormick, D. S.; Robinson, J. C.; Rodrigo, J. L., Higher order commutator estimates and local existence for the non-resistive MHD equations and related models, J. Funct. Anal., 267, 1035-1056 (2014) · Zbl 1296.35142
[17] Hmidi, T.; Keraani, S., Global solutions of the super-critical 2D quasi-geostrophic equation in Besov spaces, Adv. Math., 214, 618-638 (2007) · Zbl 1119.76070
[18] Ju, N., Existence and uniqueness of the solution to the dissipative 2D quasi-geostrophic equation in the Sobolev space, Comm. Math. Phys., 251, 365-376 (2004) · Zbl 1106.35061
[19] Ju, N., The maximum principle and the global attractor for the dissipative 2D quasi-geostrophic equation, Comm. Math. Phys., 255, 161-181 (2005) · Zbl 1088.37049
[20] Kato, T.; Ponce, G., Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41, 891-907 (1988) · Zbl 0671.35066
[21] Kiselev, A.; Nazarov, F.; Volberg, A., Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167, 445-453 (2007) · Zbl 1121.35115
[22] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248 (1934) · JFM 60.0726.05
[23] Triebel, H., Theory of Function Spaces (1983), Birkhäuser: Birkhäuser Basel · Zbl 0546.46027
[24] Wu, J., Quasi-geostrophic-type equations with initial data in Morrey spaces, Nonlinearity, 10, 1409-1420 (1997) · Zbl 0906.35078
[25] Wu, J., Dissipative quasi-geostrophic equations with \(L_p\) data, Electron. J. Differential Equations, 2001, 1-13 (2001)
[26] 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
[27] Wu, J., Existence uniqueness results for the 2-D dissipative quasi-geostrophic equation, Nonlinear Anal., 67, 3013-3036 (2007) · Zbl 1122.76014
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.