The BKM criterion to the 3D double-diffusive magneto convection systems involving planar components
Abstract
In this paper, we investigate the BKM type blowup criterion applied to 3D double-diffusive magneto convection systems. Specifically, we demonstrate that a unique local strong solution does not experience blow-up at time T, given that 
). To prove this, we employ the logarithmic Sobolev inequality in the Besov spaces with negative indices and a well-known commutator estimate established by Kato and Ponce. This result is the further improvement and extension of the previous works by O (2021) and Wu (2023).
Keyword : double-diffusive convection systems, blowup criterion, commutator estimate, regularity
How to Cite
O, C.-J., & Wu, F. (2024). The BKM criterion to the 3D double-diffusive magneto convection systems involving planar components. Mathematical Modelling and Analysis, 29(4), 684–693. https://doi.org/10.3846/mma.2024.19674
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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C.-J. O. A remark on the regularity criterion for the 3D Navier–Stokes equations in terms of two vorticity components. Nonlinear Anal. Real World Appl., 71:103840, 2023. https://doi.org/10.1016/j.nonrwa.2023.103840
M.A. Ragusa and F. Wu. Global regularity and stability of solutions to the 3D double-diffusive convection system with Navier boundary conditions. Adv. Differential Equations, 26(7/8):281–304, 2021. https://doi.org/10.57262/ade026-0708-281
N. Rudraiah. Double-diffusive magnetoconvection. Pramana, 27:233–266, 1986. https://doi.org/10.1007/BF02846340
F. Wu. Blowup criterion of strong solutions to the three-dimensional doublediffusive convection system. Bull. Malays. Math. Sci. Soc., 43(3):2673–2686, 2020. https://doi.org/10.1007/s40840-019-00828-3
F. Wu. Blowup criterion via only the middle eigenvalue of the strain tensor in anisotropic lebesgue spaces to the 3D double-diffusive convection equations. J. Math. Fluid Mech., 22:1–9, 2020. https://doi.org/10.1007/s00021-020-0483-9
F. Wu. Well-posedness and blowup criterion to the double-diffusive magnetoconvection system in 3D. Banach J. Math. Anal., 17(1):4, 2023. https://doi.org/10.1007/s43037-022-00228-z
H. Bahouri, J.Y. Chemin and R. Danchin. Fourier analysis and nonlinear partial differential equations. Springer, 2011. https://doi.org/10.1007/978-3-642-16830-7
J.T. Beale, T. Kato and A. Majda. Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Comm. Math. Phys., 94(1):61–66, 1984. https://doi.org/10.1007/BF01212349
M. Cannone. Harmonic analysis tools for solving the incompressible Navier– Stokes equations. Handbook of mathematical fluid dynamics, 3:161–244, 2004. https://doi.org/10.1016/S1874-5792(05)80006-0
B. Dong and Z. Zhang. The BKM criterion for the 3D Navier–Stokes equations via two velocity components. Nonlinear Anal. Real World Appl., 11(4):2415– 2421, 2010. https://doi.org/10.1016/j.nonrwa.2009.07.013
Z. Guo, P. Kučera and Z. Skalák. Regularity criterion for solutions to the Navier–Stokes equations in the whole 3D space based on two vorticity components. J. Math. Anal. Appl., 458(1):755–766, 2018. https://doi.org/10.1016/j.jmaa.2017.09.029
H. Huppert and D. Moore. Nonlinear double-diffusive convection. J. Fluid Mech., 78(4):821–854, 1976.
H. Huppert and J. Turner. Double-diffusive convection. J. Fluid Mech., 106:299– 329, 1981.
T. Kato and G. Ponce. Commutator estimates and the Euler and Navier–Stokes equations. Comm. Pure Appl. Math., 41(7):891–907, 1988. https://doi.org/10.1002/cpa.3160410704
C.-J. O. Regularity criterion for weak solutions to the 3D Navier–Stokes equations via two vorticity components in BMO- 1. Nonlinear Anal. Real World Appl., 59:103271, 2021. https://doi.org/10.1016/j.nonrwa.2020.103271
C.-J. O. A remark on the regularity criterion for the 3D Navier–Stokes equations in terms of two vorticity components. Nonlinear Anal. Real World Appl., 71:103840, 2023. https://doi.org/10.1016/j.nonrwa.2023.103840
M.A. Ragusa and F. Wu. Global regularity and stability of solutions to the 3D double-diffusive convection system with Navier boundary conditions. Adv. Differential Equations, 26(7/8):281–304, 2021. https://doi.org/10.57262/ade026-0708-281
N. Rudraiah. Double-diffusive magnetoconvection. Pramana, 27:233–266, 1986. https://doi.org/10.1007/BF02846340
F. Wu. Blowup criterion of strong solutions to the three-dimensional doublediffusive convection system. Bull. Malays. Math. Sci. Soc., 43(3):2673–2686, 2020. https://doi.org/10.1007/s40840-019-00828-3
F. Wu. Blowup criterion via only the middle eigenvalue of the strain tensor in anisotropic lebesgue spaces to the 3D double-diffusive convection equations. J. Math. Fluid Mech., 22:1–9, 2020. https://doi.org/10.1007/s00021-020-0483-9
F. Wu. Well-posedness and blowup criterion to the double-diffusive magnetoconvection system in 3D. Banach J. Math. Anal., 17(1):4, 2023. https://doi.org/10.1007/s43037-022-00228-z