This paper establishes the existence and uniqueness, and also presents a specific blow-up criterion, for solutions of the generalized magnetohydrodynamics (GMHD) equations in Lei–Lin spaces Xs(R3)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {X}^s(\mathbb {R}^3)$$\end{document}, by considering appropriate values for s. More precisely, if it is assumed that the initial data (u0,b0)\documentclass[12pt]{minimal}
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\begin{document}$$(u_0,b_0)$$\end{document} belong to Xs(R3)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {X}^{s}(\mathbb {R}^3)$$\end{document}, we demonstrate that there exists an instant of time T>0\documentclass[12pt]{minimal}
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\begin{document}$$T>0$$\end{document} such that (u,b)∈[CT(Xs(R3))∩LT1(Xs+2α(R3))]×[CT(Xs(R3))∩LT1(Xs+2β(R3))]\documentclass[12pt]{minimal}
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\begin{document}$$(u,b)\in [C_{T}(\mathcal {X}^s(\mathbb {R}^3))\cap L^1_{T}({\mathcal {X}}^{s+2\alpha }(\mathbb {R}^3))]\times [C_{T}(\mathcal {X}^s(\mathbb {R}^3))\cap L^1_{T}({\mathcal {X}}^{s+2\beta }(\mathbb {R}^3))]$$\end{document}, provided that α,β∈(12,1]\documentclass[12pt]{minimal}
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\begin{document}$$\alpha ,\beta \in (\frac{1}{2},1]$$\end{document} and max{α(1-2β)β,β(1-2α)α}≤s<0\documentclass[12pt]{minimal}
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\begin{document}$$\max \big \{\frac{\alpha (1-2\beta )}{\beta },\frac{\beta (1-2\alpha )}{\alpha }\big \} \le s<0$$\end{document} (here α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} and β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} are related to the fractional Laplacian that appears in the GMHD system). Furthermore, we prove that if T∗\documentclass[12pt]{minimal}
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\begin{document}$$T^*$$\end{document} (finite) is the first blow-up instant of the solution (u, b)(x, t), then limt↗T∗‖(u,b)(t)‖Xs(R3)=∞\documentclass[12pt]{minimal}
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\begin{document}$$ \lim _{t\nearrow T^*}\Vert (u,b)(t)\Vert _{\mathcal {X}^s(\mathbb {R}^3)}=\infty $$\end{document}, whether max{1-2α,1-2β,α(1-2β)β,β(1-2α)α}<s<0\documentclass[12pt]{minimal}
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\begin{document}$$\max \big \{1-2\alpha ,1-2\beta ,\frac{\alpha (1-2\beta )}{\beta },\frac{\beta (1-2\alpha )}{\alpha }\big \}< s<0$$\end{document} and α,β∈(12,1]\documentclass[12pt]{minimal}
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\begin{document}$$\alpha ,\beta \in (\frac{1}{2},1]$$\end{document}.
