Abstract
We consider the regularity criterion for the 3D MHD equations. It is proved that if the horizontal components of the velocity and magnetic fields satisfy u~,b~∈L2/(1-r)(0,T;M˙2,3/r) with 0<r<1, then the solution smooth. This improves the result given by Gala (2012).
Highlights
In this paper, we consider the following three-dimensional (3D) magnetohydrodynamic (MHD) equations: ut + (u ⋅ ∇) u − (b ⋅ ∇) b − Δu + ∇π = 0, bt + (u ⋅ ∇) b − (b ⋅ ∇) u − Δb = 0, ∇ ⋅ u = 0, (1)∇ ⋅ b = 0, u (0) = u0, b (0) = b0, where u = (u1, u2, u3) is the fluid velocity field, b = (b1, b2, b3) is the magnetic field, π is a scalar pressure, and (u0, b0) are the prescribed initial data satisfying ∇ ⋅ u0 = ∇ ⋅ b0 = 0 in the distributional sense
(1)1 reflects the conservation of momentum, (1)2 is the induction equation, and (1)3 specifies the conservation of mass
We are interested in regularity criteria involving only partial components of the velocity u, the magnetic field b, the pressure gradient ∇π, and so forth
Summary
Many sufficient conditions (see, e.g., [2,3,4,5,6,7,8,9,10,11,12,13,14] and the references therein) were derived to guarantee the regularity of the weak solution Among these results, we are interested in regularity criteria involving only partial components of the velocity u, the magnetic field b, the pressure gradient ∇π, and so forth. A measurable R3-valued pair (u, b) is called a weak solution to (1) with initial data (u0, b0), provided that the following three conditions hold:. Noticing that Ẋ r ⊂ Ṁ 2,3/r for 0 < r < 1 (see (19)), we improve the result of [16]
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