# zbMATH — the first resource for mathematics

Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation. (English) Zbl 1284.35140
This paper establishes the global in time existence of classical solutions to the two-dimensional anisotropic Boussinesq equations with vertical dissipation. When only vertical dissipation is present, there is no direct control on the horizontal derivatives and the global regularity problem is very challenging. To solve this problem, the authors bound the derivatives in terms of the $$L^{\infty}$$ -norm of the vertical velocity $$v$$ and prove that $$\parallel{v}\parallel_{L^{r}}$$ with $$2\leq{r}<\infty$$ does not grow faster than $$\sqrt{r\log{r}}$$ at any time as $$r$$ increases. A delicate interpolation inequality connecting $$\parallel{v}\parallel_{L^{\infty}}$$ and $$\parallel{v}\parallel_{L^{r}}$$ then yields the desired global regularity.

##### MSC:
 35G35 Systems of linear higher-order PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids
Full Text:
##### References:
 [1] Abidi, H.; Hmidi, T., On the global well-posedness for Boussinesq system, J. Differ. Equ., 233, 199-220, (2007) · Zbl 1111.35032 [2] Adhikari, D.; Cao, C.; Wu, J., The 2D Boussinesq equations with vertical viscosity and vertical diffusivity, J. Differ. Equ., 249, 1078-1088, (2010) · Zbl 1193.35144 [3] Adhikari, D.; Cao, C.; Wu, J., Global regularity results for the 2D Boussinesq equations with vertical dissipation, J. Differ. Equ., 251, 1637-1655, (2011) · Zbl 1232.35111 [4] Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Berlin- Heidelberg-New York, 1976 · Zbl 1143.76432 [5] Cannon, J.R., DiBenedetto, E.: The initial value problem for the Boussinesq equations with data in $$L$$\^{$$p$$}. Lecture Notes in Mathematics, Vol. 771. Springer, Berlin, pp. 129-144, 1980 · Zbl 0429.35059 [6] Cao, C.; Wu, J., Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion, Adv. Math., 226, 1803-1822, (2011) · Zbl 1213.35159 [7] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084 [8] Chae, D.; Kim, S.-K.; Nam, H.-S., Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations, Nagoya Math. J., 155, 55-80, (1999) · Zbl 0939.35150 [9] Chae, D.; Nam, H.-S., Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127, 935-946, (1997) · Zbl 0882.35096 [10] Chemin, J.-Y.: Perfect incompressible Fluids. Oxford Lecture Series in Mathematics and its Applications, Vol. 14. Oxford Science Publications, Oxford University Press, 1998 · Zbl 1154.35073 [11] Danchin, R.; Paicu, M., Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces. phys, D, 237, 1444-1460, (2008) · Zbl 1143.76432 [12] Danchin, R.; Paicu, M., Global well-posedness issues for the inviscid Boussinesq system with yudovich’s type data. commun, Math. Phys., 290, 1-14, (2009) · Zbl 1186.35157 [13] 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 [14] E, W., Engquist, B.: Blowup of solutions ot the unsteady Prandtl’s equation. Commun. Pure Appl. Math. L, 1287-1293 (1997) · Zbl 0908.35099 [15] E, W., Shu, C.: Samll-scale structures in Boussinesq convection. Phys. Fluids6, 49-58 (1994) · Zbl 0822.76087 [16] Hmidi, T.; Keraani, S., On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity. adv, Differ. Equ., 12, 461-480, (2007) · Zbl 1154.35073 [17] Hmidi, T.; Keraani, S., On the global well-posedness of the Boussinesq system with zero viscosity, Indiana Univ. Math. J., 58, 1591-1618, (2009) · Zbl 1178.35303 [18] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for a Boussinesq-Navier-Stokes system with critical dissipation, J. Differ. Equ., 249, 2147-2174, (2010) · Zbl 1200.35228 [19] Hmidi, T.; Keraani, S.; Rousset, F., Global well-posedness for Euler-Boussinesq system with critical dissipation, Commun. Partial Differ. Equ., 36, 420-445, (2011) · Zbl 1284.76089 [20] Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst., 12, 1-12, (2005) · Zbl 1274.76185 [21] Larios, A., Lunasin, E., Titi, E.S.: Global well-posedness for the 2D Boussinesq system without heat diffusion and with either anisotropic viscosity or inviscid Voigt-$$a$$ regularization. arXiv:1010.5024v1 [math.AP]. 25 Oct 2010 · Zbl 1284.35343 [22] 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 [23] Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge, 2001 · Zbl 0983.76001 [24] Majda, A.J.; Grote, M.J., Model dynamics and vertical collapse in decaying strongly stratified flows, Phys. Fluids, 9, 2932-2940, (1997) · Zbl 1185.76764 [25] Miao, C., Xue, L.: On the global well-posedness of a class of Boussinesq- Navier-Stokes systems. NoDEA. doi:10.1007/s00030-011-0114-5 · Zbl 1235.76020 [26] Moffatt, H.K.: Some remarks on topological fluid mechanics. In: Ricca, R.L. (ed.) An Introduction to the Geometry and Topology of Fluid Flows, pp. 3-10. Kluwer, Dordrecht, 2001 · Zbl 1100.76500 [27] Pedlosky, J.: Geophysical Fluid Dyanmics. Springer, New York, 1987 · Zbl 0713.76005 [28] Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations. Walter de Gruyter, Berlin, New York, 1996 · Zbl 0873.35001 [29] Triebel, H.: Theory of Function Spaces II. Birkhauser Verlag, 1992 · Zbl 0763.46025
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.