×

zbMATH — the first resource for mathematics

Global existence results for the anisotropic Boussinesq system in dimension two. (English) Zbl 1223.35249
This paper is a step forward concerning the analysis of a Boussinesq system in 2D, when the diffusivity and viscosity are variable only in horizontal direction and act in only one equation. This model is related with geophysical flows. In contrast with previous papers in the field, the present results are concerning the use of a vertical buoyancy force. Global weak solutions are considered, and existence and uniqueness theorems are obtained, assuming initial velocity in \(H^1\) and initial temperature in \(L^2\), by using the Friedrichs method. Some very interesting inequalities given in the appendix are used to get a priori estimates of the solution.

MSC:
35Q30 Navier-Stokes equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35D30 Weak solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76U05 General theory of rotating fluids
86A04 General questions in geophysics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1016/j.jde.2006.10.008 · Zbl 1111.35032 · doi:10.1016/j.jde.2006.10.008
[2] Aubin J.-P., C.-R. Acad. Sci. Paris 256 pp 5042–
[3] DOI: 10.1007/BF00377659 · Zbl 0821.76012 · doi:10.1007/BF00377659
[4] DOI: 10.1007/978-3-642-16830-7 · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7
[5] Bony J.-M., Ann. Sci.École Norm. Sup. 14 pp 209– · Zbl 0495.35024 · doi:10.24033/asens.1404
[6] DOI: 10.1007/BFb0086903 · doi:10.1007/BFb0086903
[7] DOI: 10.1016/j.aim.2005.05.001 · Zbl 1100.35084 · doi:10.1016/j.aim.2005.05.001
[8] J.Y. Chemin, Fluides Parfaits Incompressibles (Astérisque, 1995) p. 230.
[9] Chemin J.-Y., J. Anal. Math. 77 pp 25–
[10] DOI: 10.1051/m2an:2000143 · Zbl 0954.76012 · doi:10.1051/m2an:2000143
[11] Chemin J.-Y., Mathematical Geophysics: An Introduction to Rotating Fluids and to the Navier–Stokes Equations (2006) · Zbl 1205.86001
[12] Danchin R., Rev. Mat. Iber. 21 pp 863–
[13] DOI: 10.1070/RM2007v062n03ABEH004412 · Zbl 1139.76011 · doi:10.1070/RM2007v062n03ABEH004412
[14] Danchin R., Bull. Soc. Math.France 136 pp 261– · Zbl 1162.35063 · doi:10.24033/bsmf.2557
[15] DOI: 10.1016/j.physd.2008.03.034 · Zbl 1143.76432 · doi:10.1016/j.physd.2008.03.034
[16] DOI: 10.1007/s00220-009-0821-5 · Zbl 1186.35157 · doi:10.1007/s00220-009-0821-5
[17] DOI: 10.1007/BF01393835 · Zbl 0696.34049 · doi:10.1007/BF01393835
[18] Weinan E., Phys. Fluids 6 pp 49–
[19] P. Gérard, Sém. Bourbaki 92, eds. J.Y. Chemin and J.M. Delort (Astérisque, 1992) pp. 411–444.
[20] DOI: 10.1007/BF02006004 · Zbl 0681.76048 · doi:10.1007/BF02006004
[21] Hmidi T., Adv. Diff. Eqns. 12 pp 461–
[22] DOI: 10.1512/iumj.2009.58.3590 · Zbl 1178.35303 · doi:10.1512/iumj.2009.58.3590
[23] Hou T., Disc. Cont. Dynam. Syst. 12 pp 1–
[24] DOI: 10.1137/S0036141000382126 · Zbl 1011.35105 · doi:10.1137/S0036141000382126
[25] DOI: 10.1007/BF02547354 · JFM 60.0726.05 · doi:10.1007/BF02547354
[26] Paicu M., Rev. Mat. Iber. 21 pp 179–
[27] DOI: 10.1007/978-1-4612-4650-3 · doi:10.1007/978-1-4612-4650-3
[28] Yudovich V., Akad. Nauk SSSR. Žurnal Vyčislitel’noĭ Mat. Mat. Fiz. 3 pp 1032–
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.