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In this paper we consider an initial value problem of the 2D MHD equations with velocity dissipation and without magnetic diffusion. We establish a new magnetic regularity criterion in terms of the magnetic field. In contrast to the magnetic regularity criterion ∇b∈L1(0,T;BMO(R2))$\nabla b\in L^{1}(0,T; BMO(\mathbb {R}^{2}))$, our regularity criterion ∫0T(∥b⊗b(s)∥B∞,10(R2)+∥b⊗b(s)∥L2(R2))ds<∞$\int_{0}^{T} (\Vert b\otimes b(s)\Vert _{B_{\infty,1}^{0}(\mathbb {R}^{2})} +\Vert b\otimes b(s)\Vert _{L^{2}(\mathbb {R}^{2})} )\,ds<\infty$ is different; for example, our simplified regularity criterion ∫0T∥b(s)∥B∞,1ε(R2)2ds<∞$\int_{0}^{T}\Vert b(s)\Vert ^{2}_{B_{\infty,1}^{\varepsilon}(\mathbb {R}^{2})}\,ds<\infty$ requires higher time integrability and lower regularity of space.

A magnetic regularity criterion for the 2D MHD equations with velocity dissipation