We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled Lx, tp, q-norm of the velocity with 1 ≤ 3 / p + 2 / q ≤ 2, 1 ≤ q ≤ ∞ is sufficiently small near z and if the scaled Lx, tl, m-norm of the magnetic field with 1 ≤ 3 / l + 2 / m ≤ 2, 1 ≤ m ≤ ∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations.
Bibliographical noteFunding Information:
The authors wish to express their sincere gratitude to Professor Tai-Peng Tsai for careful reading of the manuscript and many useful comments. The first author was supported partially by KRF-2006-311-C00007 and KOSEF-R01-2008-000-11008-0. The second author was supported partially by KRF-2006-311-C00208 and KOSEF-F01-2006-000-10075-0.
All Science Journal Classification (ASJC) codes
- Applied Mathematics