On Two-Dimensional Magnetic Bénard Problem with Mixed Partial Viscosity
Tóm tắt
In this paper, we deal with the Cauchy problem of the two-dimensional magnetic Bénard problem with mixed partial viscosity. More precisely, the global well-posedness of 2D magnetic Bénard problem without thermal diffusivity and with vertical or horizontal magnetic diffusion is obtained. Moreover, the global regularity and some conditional regularity of strong solutions are obtained for 2D magnetic Bénard problem with mixed partial viscosity. The results extend the recent
work (Appl Math Letter 26:627–630, 2013) on the global regularity of the magnetic Bénard problem with full dissipation and magnetic diffusion in two dimensions.
Tài liệu tham khảo
Adhikari D., Cao C.S., Wu J.H.: Global regularity results for the 2D Boussinesq equations with vertical dissipation. J. Diff. Equ. 251, 1637–1655 (2011)
Ambrosetti, A., Prodi, G.: A primer of nonlinear analysis, vol. 34. Cambridge Stud. Adv. Math. (1995)
Cao C.S., Regmi D., Wu J.H.: The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion. J. Differ. Equ. 254, 2661–2681 (2013)
Cao C.S., Wu J.H.: Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion. Adv. Math. 226, 1803–1822 (2011)
Cao C.S., Wu J.H.: Global regularity for the two-dimensional anisotropic Boussinesq equations with vertical dissipation. Arch. Ration. Mech. Anal. 208(3), 985–1004 (2013)
Chae D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203(2), 497–513 (2006)
Chandrasekhar S.: Hydrodynamic and hydromagnetic stability. The international series of monographs on physics. Clarendon Press, Oxford (1961)
Chemin J.-Y., Desjardins B., Gallagher I., Grenier E.: Fluids with anisotropic viscosity. Math. Model. Numer. Anal. 34(2), 315–335 (2000)
Chemin J.-Y., Desjardins B., Gallagher I., Grenier E.: Mathematical geophysics: an introduction to rotating fluids and the Navier-Stokes equations. Oxford University Press, Oxford (2006)
Danchin R., Paicu M.: Global existence results for the anisotropic Boussinesq system in dimension two. Math. Models Methods Appl. Sci. 21, 421–457 (2011)
Du L.L., Zhou D.Q.: Global well-posedness of 2D magnetohydrodynamic flows with partial dissipation and magnetic diffusion. SIAM. J. Math. Anal. 47(2), 1562–1589 (2015)
Duvaut G., Lions J.-L.: Inéquations en thermoélasticité et magnétohydrodynamique. Arch. Ration. Mech. Anal. 46, 241–279 (1972)
Guo B.L.: Spectral method for solving two-dimensional Newton-Boussinesq equation. Acta Mathematicae Appication Sinica. 5(3), 208–218 (1989)
Fan J.S., Ozawa T.: Regularity criteria for the 2D MHD system with horizontal dissipation and horizontal magnetic diffusion. Kinet. Relat. Models. 7, 45–56 (2014)
Hou T.Y., Li C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12, 1–12 (2005)
Iftimie D.: A uniqueness result for the Navier-Stokes equations with vanishing vertical viscosity. SIAM J. Math. Anal. 33(6), 1483–1493 (2002)
Kozono H.: Weak and classical solutions of the two-dimensional magnetohydrodynamic equations. Tohoku Math. J. 41, 471–488 (1989)
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)
Lin H.X., Du L.L.: Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions. Nonlinearity 26, 219–239 (2013)
Majda A.J., Bertozzi A.L.: Vorticity and incompressible flow. Cambridge texts in applied mathematics, vol. 27. Cambridge University Press, Cambridge (2002)
Ma T., Wang S.H.: Rayleigh-Bénard convection: dynamics and structure in the physical space. Commun. Math. Sci. 5(3), 553–574 (2007)
Miao C.X., Zheng X.X.: On the global well-posedness for the Boussinesq system with horizontal dissipation. Commun. Math. Phys. 321, 33–67 (2013)
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 Academic Publishers, Dordrecht (2001)
Mulone G., Rionero S.: Necessary and sufficient conditions for nonlinear stability in the magnetic Bénard problem. Arch. Ration. Mech. Anal. 166, 197–218 (2003)
Nakamura M.A.: On the magnetic Bénard problem. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 38, 359–393 (1991)
Paicu M.: Équation anisotrope de Navier-Stokes dans des espaces critiques. Revista Matemática Iberoamericana 21(1), 179–235 (2005)
Rabinowitz P.H.: Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Ration. Mech. Anal. 29, 32–57 (1968)
Sermange M., Temam R.: Some mathematical questions related to the MHD equations. Commun. Pure. Appl. Math. 36, 635–664 (1983)
Wu J.H.: Global regularity for a class of generalized magnetohydrodynamic equations. J. Math. Fluid Mech. 13, 295–305 (2011)
Wu G., Xue L.T.: Global well-posedness for the 2D inviscid Bénard system with fractional diffusivity and Yudovich’s type data. J. Differ. Equ. 253, 100–125 (2012)
Wu G., Zheng X.X.: Global well-posedness for the two-dimensional nonlinear Boussinesq equations with vertical dissipation. J. Differ. Equ. 255, 2891–2926 (2013)
Zhou, Y., Fan, J.S., Nakamura, G.: Global Cauchy problem for a 2D magnetic Bénard problem with zero thermal conducivity. Appl. Math. Lett. 26, 627–630 (2013)