Boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations

Abstract

We present some new regularity criteria for suitable weak solutions of magnetohydrodynamic equations near boundary in dimension three. We prove that suitable weak solutions are Hölder continuous near boundary provided that either the scaled L p,q x,t -norm of the velocity with 3/p + 2/q ≤ 2, 2 < q < ∞, or the scaled L p,q x,t -norm of the vorticity with 3/p + 2/q ≤ 3, 2 < q < ∞ are sufficiently small near the boundary.

Regularity criteria for suitable weak solutions to the four dimensional incompressible magneto-hydrodynamic equations near boundary

“…We remark that by using the same method we can get an alternative proof of Kang, Kim's results [15] without using any compactness argument and our method also provides a different approach than [12] to prove interior partial regularity results. It remains an interesting open problem whether a similar result can be obtained for higher dimensional MHD equations (d ≥ 5 for the time-dependent case).…”

“…The boundary conditions of u and H are given as following: u = 0, and H · ν = 0, ∇H · ν = 0, ∀ x ∈ ∂Ω, (1.2) where ν is the outward unit normal vector along the boundary ∂Ω. The boundary condition for H is equivalent to the slip-condition described in [15] in three dimension. The MHD equations usually describe the dynamics of the interaction of moving conducting fluids with electro-magnetic fields which are frequently observed in nature and industry, e.g., plasma liquid metals, gases (see [5,10]).…”

“…is satisfied for some ε near a interior point z 0 , then z 0 is regular. Kang and Kim extended this kind criteria to the boundary case in [15]. The main idea of these works is that the terms induced by Π and H · ∇H in the local energy inequality (1.3) all contain the velocity u, if some scaled norm of u is small, then these terms can be controlled by timing small parameter on scale-invariant quantities of the pressure and using an iteration method.…”

“…Remark 1.3. Motivated by [15,34], we proved this ε-regularity criteria, which only requires that the scaled norm of the velocity is small.…”

“…Compared to the Navier-Stokes equations ( [8,9]), the main difference in the MHD equations is to control the pressure Π and the nonlinear term H · ∇H. As we explained before, motivated by [15,34], we can control the terms induced by Π and H · ∇H in the local energy inequality (1.3) with the assumption of Theorem 1.2. However, Theorem 1.2 can not infer the optimal partial regularity result.…”