Acknowledgement
This work is supported by the National Key R&D Program of China (2021YFA1000800) and the K.C.Wong Education Foundation, Chinese Academy of Sciences.
References
- B. Dong, J. Li and J. Wu, Global well-posedness and large-time decay for the 2D micropolar equations, J. Differential Equations, 262 (2017), 3488-3523. https://doi.org/10.1016/j.jde.2016.11.029
- B. Dong, J. Wu, X. Xu and Z. Ye, Global regularity for the 2D micropolar equations with fractional dissipation, Discrete Contin. Dyn. Syst., 38 (2018), 4133-4162. https://doi.org/10.3934/dcds.2018180
- D. Wang, J. Wu and Z. Ye, Global regularity of the three-dimensional fractional micropolar equations, J. Math. Fluid Mech., 22 (2020), 36 pp. https://doi.org/10.1007/s00021-020-00499-2
- D. Chae, Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203 (2006), 497-515. https://doi.org/10.1016/j.aim.2005.05.001
- D. Chae and J. Wu, The 2D Boussinesq equations with logarithmically supercritical velocities, Adv. Math., 230 (2012), 1618-1645. https://doi.org/10.1016/j.aim.2012.04.004
- Z. Ye, A logarithmically improved regularity criterion of smooth solutions for the 3D Boussinesq equations, Osaka J. Math., 53 (2016), 417-423.
- M. E. Schonbek, L2 decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 88 (1985), 209-222. https://doi.org/10.1007/BF00752111
- R. H. Guterres, J. R. Nunes and C. F. Perusato, On the large time decay of global solutions for the micropolar dynamics in L2(Rn), Nonlinear Anal. Real World Appl., 45 (2019), 789-798. https://doi.org/10.1016/j.nonrwa.2018.08.002
- Y. Guo, Y. Jia and B. Dong, Time decay rates of the micropolar equations with zero angular viscosity, Bull. Malays. Math. Sci. Soc., 44 (2021), 3663-3675. https://doi.org/10.1007/s40840-021-01138-3
- H. Ye, Q. Wang and Y. Jia, Well-posedness and large time decay for the 3D micropolar equations with only velocity dissipation, Nonlinear Anal., 219 (2022), Paper No. 112796, 25 pp.
- L. Brandolese and M. E. Schonbek, Large time decay and growth for solutions of a viscous Boussinesq system, Trans. Amer. Math. Soc., 364 (2012), 5057-5090. https://doi.org/10.1090/S0002-9947-2012-05432-8
- H. O. Bae and H. J. Choe, Decay rate for the incompressible flows in half spaces, Math. Z., 238 (2001), 799-816. https://doi.org/10.1007/s002090100276
- H. O. Bae and B. J. Jin, Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations, J. Differential Equations, 209 (2005), 365-391. https://doi.org/10.1016/j.jde.2004.09.011
- H. O. Bae and B. J. Jin, Asymptotic behavior for the Navier-Stokes equations in 2D exterior domains, J. Funct. Anal., 240 (2006), 508-529. https://doi.org/10.1016/j.jfa.2006.04.029
- H. O. Bae and B. J. Jin, Temporal and spatial decay rates of Navier-Stokes solutions in exterior domains, Bull. Korean Math. Soc., 44 (2007), 547-567. https://doi.org/10.4134/BKMS.2007.44.3.547
- C. He and D. Zhou, Existence and asymptotic behavior for an incompressible Newtonian flow with intrinsic degree of freedom, Math. Methods Appl. Sci., 37 (2014), 1191-1205. https://doi.org/10.1002/mma.2880
- T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. https://doi.org/10.1002/cpa.3160410704
- H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations, Academic Press, New York, 1989.
- H. Sohr, The Navier-Stokes equations: an elementary functional analytic approach, Birkhauser Advanced Texts, 2001.
- R. Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.