Stability of the 2D Boussinesq system with partial dissipation. (English) Zbl 1493.35083
Summary: This paper establishes the global stability for the 2D Boussinesq system with partial dissipation and horizontal thermal diffusion. When there is no thermal diffusion, the stability of the temperature gradient remains an open problem. We extend the \(H^1\)-stability in [7] to \(H^2\)-stability which we care about and obtain the large time behavior of the linearized system.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B35 | Stability in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 76R10 | Free convection |
| 80A19 | Diffusive and convective heat and mass transfer, heat flow |
Keywords:
Boussinesq equation; hydrostatic equilibrium; partial dissipation; large-time behavior; stabilityReferences:
| [1] | Biswas, A.; Foias, C.; Larios, A., On the attractor for the semi-dissipative Boussinesq equations, Ann. Inst. H. Poincare Anal. Non Lineair, 34, 381-405 (2017) · Zbl 1361.35138 · doi:10.1016/j.anihpc.2015.12.006 |
| [2] | Bahouri, H.; Chemin, J-Y; Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations (2011), Berlin: Springer, Berlin · Zbl 1227.35004 · doi:10.1007/978-3-642-16830-7 |
| [3] | 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 · doi:10.1016/j.aim.2010.08.017 |
| [4] | Castro, A., Córdoba, D. Lear, D.: On the asymptotic stability of stratified solutions for the 2D Boussinesq equations with a velocity damping term. arXiv:1805.05179v2[math.AP] 1, Oct (2018) · Zbl 1425.35149 |
| [5] | Davidson, PA, An Introduction to Magnetohydrodynamics (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 0974.76002 · doi:10.1017/CBO9780511626333 |
| [6] | Doering, CR; Wu, J.; Zhao, K.; Zheng, X., Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion, Phys. D, 376/377, 144-159 (2018) · Zbl 1398.35164 · doi:10.1016/j.physd.2017.12.013 |
| [7] | Ji, R.; Li, D.; Wei, Y.; Wu, J., Stability of hydrostatic equilibrium to the 2D Boussinesq systems with partial dissipation, Appl. Math. Lett., 98, 392-397 (2019) · Zbl 1448.35397 · doi:10.1016/j.aml.2019.06.019 |
| [8] | Lai, M.; Pan, R.; Zhao, K., Initial boundary value problem for 2D viscous Boussinesq equations, Arch. Ration. Mech. Anal., 199, 111-124 (2011) · Zbl 1231.35171 · doi:10.1007/s00205-010-0357-z |
| [9] | Lin, H.; Du, L., Regularity criteria for incompressible magnetohydrodynamics equations in three dimensions, Nonlinearity, 26, 219-239 (2013) · Zbl 1273.35076 · doi:10.1088/0951-7715/26/1/219 |
| [10] | Pedlosky, J., Geophysical Fluid Dyanmics (1987), New York: Springer, New York · Zbl 0713.76005 · doi:10.1007/978-1-4612-4650-3 |
| [11] | Tao, L.; Wu, J.; Zhao, K.; Zheng, X., Stability near hydrostatic equilibrium to the 2D Boussinesq equations without thermal diffusion, Arch. Ration. Mech. Anal., 237, 585-630 (2020) · Zbl 1437.35549 · doi:10.1007/s00205-020-01515-5 |
| [12] | White, FM, Fluid Mechanics (2008), New York: McGraw-Hill, New York |
| [13] | Wan, R., Global well-posedness for the 2D Boussinesq equations with a velocity damping term, Discrete Contin Dyn Syst, 39, 5, 2709-2730 (2019) · Zbl 1412.35269 · doi:10.3934/dcds.2019113 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
