The weak sigma-attractor for the semi-dissipative 2D Boussinesq system

He, Jinfang; Sun, Chunyou

doi:10.1090/proc/14807

The weak sigma-attractor for the semi-dissipative 2D Boussinesq system
HTML articles powered by AMS MathViewer

by Jinfang He and Chunyou Sun PDF

Proc. Amer. Math. Soc. 148 (2020), 1219-1231 Request permission

Abstract:

In this paper, we obtain the attraction properties for the velocity variable of the 2D Boussinesq equations with viscosity and without heat diffusion in the sense of the strong topology of $V$ and prove that the weak sigma-attractor has a pancake-like structure, which answer partly some questions arising in Biswas et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), pp. 381–405] and enrich the structure of the weak sigma-attractor.

References

Animikh Biswas, Ciprian Foias, and Adam Larios, On the attractor for the semi-dissipative Boussinesq equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 34 (2017), no. 2, 381–405. MR 3610937, DOI 10.1016/j.anihpc.2015.12.006
Dongho Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math. 203 (2006), no. 2, 497–513. MR 2227730, DOI 10.1016/j.aim.2005.05.001
Peter Constantin and Ciprian Foias, Navier-Stokes equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. MR 972259
Raphaël Danchin and Marius Paicu, Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France 136 (2008), no. 2, 261–309 (French, with English and French summaries). MR 2415344, DOI 10.24033/bsmf.2557
Charles R. Doering, Jiahong Wu, Kun Zhao, and Xiaoming Zheng, Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion, Phys. D 376/377 (2018), 144–159. MR 3815212, DOI 10.1016/j.physd.2017.12.013
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
C. Foias, O. Manley, and R. Temam, Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal. 11 (1987), no. 8, 939–967. MR 903787, DOI 10.1016/0362-546X(87)90061-7
Jack K. Hale, Asymptotic behavior of dissipative systems, Mathematical Surveys and Monographs, vol. 25, American Mathematical Society, Providence, RI, 1988. MR 941371, DOI 10.1090/surv/025
Taoufik Hmidi and Sahbi Keraani, On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity, Adv. Differential Equations 12 (2007), no. 4, 461–480. MR 2305876
Weiwei Hu, Igor Kukavica, and Mohammed Ziane, Persistence of regularity for the viscous Boussinesq equations with zero diffusivity, Asymptot. Anal. 91 (2015), no. 2, 111–124. MR 3305763, DOI 10.3233/asy-141261
Thomas Y. Hou and Congming Li, Global well-posedness of the viscous Boussinesq equations, Discrete Contin. Dyn. Syst. 12 (2005), no. 1, 1–12. MR 2121245, DOI 10.3934/dcds.2005.12.1
Adam Larios, Evelyn Lunasin, and Edriss S. Titi, Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion, J. Differential Equations 255 (2013), no. 9, 2636–2654. MR 3090072, DOI 10.1016/j.jde.2013.07.011
Ming-Jun Lai, Ronghua Pan, and Kun Zhao, Initial boundary value problem for two-dimensional viscous Boussinesq equations, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 739–760. MR 2771665, DOI 10.1007/s00205-010-0357-z
Songsong Lu, Hongqing Wu, and Chengkui Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 701–719. MR 2153139, DOI 10.3934/dcds.2005.13.701
Andrew Majda, Introduction to PDEs and waves for the atmosphere and ocean, Courant Lecture Notes in Mathematics, vol. 9, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. MR 1965452, DOI 10.1090/cln/009
Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
Qingfeng Ma, Shouhong Wang, and Chengkui Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J. 51 (2002), no. 6, 1541–1559. MR 1948459, DOI 10.1512/iumj.2002.51.2255
J.Pedlosky, Geophysical Fluid Dynamics, Springer, New York, 1987.
James C. Robinson, Infinite-dimensional dynamical systems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2001. An introduction to dissipative parabolic PDEs and the theory of global attractors. MR 1881888, DOI 10.1007/978-94-010-0732-0
Roger Temam, Navier-Stokes equations and nonlinear functional analysis, 2nd ed., CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 66, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1995. MR 1318914, DOI 10.1137/1.9781611970050
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312, DOI 10.1007/978-1-4612-0645-3