Abstract
We consider the well-posedness and blowup criterion to the double-diffusive magnetoconvection system in 3D. First, we establish the existence and uniqueness of the local strong solution to the system (1.1) in \(H^1({\mathbb {R}}^3)\) with arbitrary initial data, and obtain the global strong solution when the \(L^2\) norm of the initial data is small. This result can be regarded as a generalization of the methods in Chen et al. (J Math Phys 60(1):011511, 2019). Then, we provide some sufficient conditions for the breakdown of local strong solution to system (1.1) in terms of velocity (or gradient of velocity) in weak \(L^p\) spaces. Finally, we focus on blowup criterion only depends on partial derivative of the planar components \(({\tilde{u}}, {\tilde{b}})\) without \(u_3\) and \(b_3\) in \({{\text {BMO}}}^{-1}\) space. More precisely, if local solution satisfies
then the strong solution \((u, b, \theta , s)\) can be extended smoothly beyond \(t=T\). This improves and extends several previous BKM’s criteria (Chol-Jun in Nonlinear Anal Real World Appl 59:103271, 2021; Guo et al. in J Math Anal Appl 458(1):755–766, 2018).
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The author would like to express the heartfelt thanks to the anonymous referees and the editor for their constructive comments and helpful suggestions that have contributed to the final preparation of the paper. The author would like to thank sincerely Dr. Zhengmao Chen for helpful comments on the revise version of the present paper.
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Wu, F. Well-posedness and blowup criterion to the double-diffusive magnetoconvection system in 3D. Banach J. Math. Anal. 17, 4 (2023). https://doi.org/10.1007/s43037-022-00228-z
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DOI: https://doi.org/10.1007/s43037-022-00228-z