Abstract
New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first result for global boundedness of the solution \({(u, \theta)}\) in \({D(A)\times H^1}\) improves considerably the main result of the recent article (Hu et al. in J Math Phys 54(8):081507, 2013). Our second result on global boundedness of the solution \({(u, \theta)}\) in \({V\times H^1}\) for both bounded domain and the whole space \({\mathbb{R}^{2}}\) is a new one. It has been open and also seems much more challenging than the first result. Global regularity of the solution \({(u, \theta)}\) in \({D(A)\times H^{2}}\) is also proved.
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References
Brezis H., Gallouet T.: Nonlinear Schrödinger evolution equations. Nonlinear Anal. Theory Methods Appl. 4(4), 677–681 (1980)
Chae D.: Global regularity for the 2D Boussinesq equations with partial viscosity terms. Adv. Math. 203(2), 497–513 (2006)
Constantin, P., Foias, C.: Navier–Stokes Equations. University of Chicago Press, Chicago, IL (1988)
Danchin R., Paicu M.: Le théorème de Leray et le théorème de Fujita-Kato pour le système de Boussinesq partiellement visqueux. Bull. Soc. Math. France 136(2), 261–309 (2008)
Danchin R., Paicu M.: Global existence results for the anisotropic Boussinesq system in dimension two. Math. Models Meth. Appl. Sci. 21(3), 421–457 (2011)
He L.: Smoothing estimates of 2d incompressible Navier–Stokes equations in bounded domains with applications. J. Funct. Anal. 262(7), 3430–3464 (2012)
Hu, W., Kukavica, I., Ziane, M.: On the regularity for the Boussinesq equations in a bounded domain. J. Math. Phys. 54(8), 081507, 10 (2013)
Hou T.Y., Li C.: Global well-posedness of the viscous Boussinesq equations. Discrete Contin. Dyn. Syst. 12(1), 1–12 (2005)
Lai M.J., Pan R., Zhao K.: Initial boundary value problem for two-dimensional viscous Boussinesq equations. Arch. Ration. Mech. Anal. 199(3), 739–760 (2011)
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(9), 2636–2654 (2013)
Li Y.C.: Global regularity for the viscous Boussinesq equations. Math. Methods Appl. Sci. 27(3), 363–369 (2004)
Temam, R.: Navier–Stokes equations. Theory and numerical analysis, In: Studies in Mathematics and its Applications, vol. 2. North-Holland (1977) (reprinted with corrections by AMS, 2001)
Temam, R.: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences Series, vol. 68. Spring-Verlag, New York (1988) (second augmented edition, 1997)
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Ju, N. Global Regularity and Long-time Behavior of the Solutions to the 2D Boussinesq Equations without Diffusivity in a Bounded Domain. J. Math. Fluid Mech. 19, 105–121 (2017). https://doi.org/10.1007/s00021-016-0277-2
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DOI: https://doi.org/10.1007/s00021-016-0277-2