Well-posedness and blowup criterion to the double-diffusive magnetoconvection system in 3D

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

$$\begin{aligned} \int ^{T}_{0}\Vert \nabla _{h}{\tilde{u}}(t)\Vert ^{2}_{{\text {BMO}}^{-1}} +\Vert \nabla _{h}{\tilde{b}}(t)\Vert ^{2}_{{\text {BMO}}^{-1}} {\text {d}}t<\infty , \end{aligned}$$

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).