AbstractIn this paper, we investigate an initial boundary value problem for two-dimensional inhomogeneous incompressible MHD system with density-dependent viscosity. First, we establish a blow-up criterion for strong solutions with vacuum. Precisely, the strong solution exists globally if $\|\nabla \mu (\rho )\|_{L^{\infty }(0, T; L^{p})}$
∥
∇
μ
(
ρ
)
∥
L
∞
(
0
,
T
;
L
p
)
is bounded. Second, we prove the strong solution exists globally (in time) only if $\|\nabla \mu (\rho _{0})\|_{L^{p}}$
∥
∇
μ
(
ρ
0
)
∥
L
p
is suitably small, even the presence of vacuum is permitted.
Blow-up Criteria of Strong Solutions to the Ericksen-Leslie System in ℝ3