Abstract
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 $\nabla b\in L^{1}(0,T; BMO(\mathbb {R}^{2}))$ , our regularity criterion $\int_{0}^{T} (\Vert b\otimes b(s)\Vert _{B_{\infty,1}^{0}(\mathbb {R}^{2})} +\Vert b\otimes b(s)\Vert _{L^{2}(\mathbb {R}^{2})} )\,ds<\infty$ is different; for example, our simplified regularity criterion $\int_{0}^{T}\Vert b(s)\Vert ^{2}_{B_{\infty,1}^{\varepsilon}(\mathbb {R}^{2})}\,ds<\infty$ requires higher time integrability and lower regularity of space.
Highlights
In this paper we consider the global regularity on the D incompressible magnetohydrodynamic (MHD) equations with velocity dissipation and without magnetic diffusion, ⎧ ⎪⎪⎪⎨∂∂tt u b + + u u · · ∇u ∇b + = ∇p = b b · ∇u, ·∇b + u, x ∈ R, t
3 Proof of Theorem 1.1 we prove our main result Theorem
∇u(s) L∞ ds, we obtain the global bounds of w H and j H
Summary
In this paper we consider the global regularity on the D incompressible magnetohydrodynamic (MHD) equations with velocity dissipation and without magnetic diffusion,. Jiu and Liu [ ] discussed the global regularity for the D axisymmetric MHD equations with horizontal dissipation and vertical magnetic diffusion. Due to lack of the magnetic diffusion, it is very difficult to get global estimates of the local solution in any Sobolev spaces. Two kinds of the coupled space-time Besov spaces LrT Bsp,q and LrT Bsp,q (r ≥ ) are defined, respectively, as follows: LrT Bsp,q = u ∈ S Rd , u LrT Bsp,q(Rd) := js ju Lp lq LrT < ∞ , LrT Bsp,q = u ∈ S Rd , u LrT Bsp,q(Rd) := js ju LrT Lp lq < ∞. There is a constant such that for all k ∈ N ∪ { }, ≤ p ≤ q ≤ ∞, and f ∈ Lp, we have supp f ⊂ λB ⇒
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