zbMATH — the first resource for mathematics

Global well-posedness of the 2D Boussinesq equations with vertical dissipation. (English) Zbl 1336.35297
Summary: We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data \((u_0,\theta_0)\) are required to be only in the space \(X=\{f\in L^2(\mathbb R^2)|\partial_x f \in L^2(\mathbb R^2)\}\), and thus our result generalizes that of C. Cao and J. Wu [Arch. Ration. Mech. Anal. 208, No. 3, 985–1004 (2013; Zbl 1284.35140)], where the initial data are assumed to be in \(H^2(\mathbb R^2)\). The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the \(L^\infty(\mathbb R^2)\) norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.

35Q35 PDEs in connection with fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text: DOI arXiv
[1] Brézis, H.; Gallouet, T., Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4, 677-681, (1980) · Zbl 0451.35023
[2] Brézis, H.; Wainger, S., A note on limiting cases of Sobolev embeddings and convolution inequalities, Commun. Partial Diff. Equ., 5, 773-789, (1980) · Zbl 0437.35071
[3] Cao, C.; Farhat, A.; Titi, E.S., Global well-posedness of an inviscid three-dimensional pseudo-hasegawa-mima model, Commun. Math. Phys., 319, 195-229, (2013) · Zbl 1308.35200
[4] Cao C., Li J., Titi E.S.: Global well-posedness for the 3D primitive equations with only horizontal viscosity and diffusion. Commun. Pure Appl. Math. doi:10.1002/cpa.21576 · Zbl 1351.35125
[5] Cao C., Li J., Titi E.S.: Strong solutions to the 3D primitive equations with horizontal dissipation: near \(H\)\^{1} initial data (preprint) · Zbl 1366.35123
[6] Cao, C.; Wu, J., Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation, Arch. Rational Mech. Anal., 208, 985-1004, (2013) · Zbl 1284.35140
[7] Cannon J.R., DiBenedetto E.: The initial value problem for the Boussinesq equations with data in Lp. In: Approximation Methods for Navier-Stokes Problems, Proc. Sympos., Univ. Paderborn, Paderborn, 1979, Lecture Notes in Math., vol. 771. Springer, Berlin, pp. 129-144 1980 · Zbl 0451.35023
[8] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084
[9] Constantin P., Foias C.: Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, 1988 · Zbl 0646.76098
[10] Danchin, R.; Paicu, M., LES théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, 136, 261-309, (2008) · Zbl 1162.35063
[11] Danchin, R.; Paicu, M., Global existence results for the anisotropic Boussinesq system in dimension two, Math. Models Methods Appl. Sci., 21, 421-457, (2011) · Zbl 1223.35249
[12] Foias, C.; Manley, O.; Temam, R., Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal. Theory Methods Appl., 11, 939-967, (1987) · Zbl 0646.76098
[13] Hmidi, T.; Keraani, S., On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Diff. Equ., 12, 461-480, (2007) · Zbl 1154.35073
[14] Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Sys., 12, 1-12, (2005) · Zbl 1274.76185
[15] Hu W., Kukavica I., Ziane M.: On the regularity for the Boussinesq equations in a bounded domain. J. Math. Phys. 54(8), 081507, 10 pp. (2013) · Zbl 1293.35246
[16] Lai, M.; Pan, R.; Zhao, K., Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199, 739-760, (2011) · Zbl 1231.35171
[17] Larios, A.; Lunasin, E.; Titi, E.S., Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differ. Equ., 255, 2636-2654, (2013) · Zbl 1284.35343
[18] Lieb E.H., Loss M.: Analysis Second edition., Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, 2001 · Zbl 0966.26002
[19] Majda A.J.: Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lecture Notes in Mathematics, vol. 9. AMS/CIMS, 2003 · Zbl 1278.76004
[20] Majda A.J., Bertozzi A.L.: Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, 2001 · Zbl 0983.76001
[21] Pedlosky J.: Geophysical Fluid Dynamics, Spring, New York, 1987. · Zbl 0713.76005
[22] Temam R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn, Appl. Math. Sci., vol. 68, Springer-Verlag, New York, 1997 · Zbl 0871.35001
[23] Vallis G.K.: Atmospheric and Oceanic Fluid Dynamics, Cambridge Univ. Press, 2006 · Zbl 1374.86002
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.