In this paper, we investigate the Cauchy problem for Cahn–Hilliard equation with inertial term in n-dimensional space. Based on the decay estimate of solutions to the corresponding linear equation, we define a solution space. Under small condition on the initial value, we prove the global existence and asymptotic behavior of the solution in the corresponding Sobolev spaces by the contraction mapping principle.

In this paper, we derive a blow-up criterion of smooth solutions to the incompressible magneto-micropolar fluid equations with partial viscosity in two space dimensions. Our proof is based on careful Hölder estimates of heat and transport equations and the standard Littlewood-Paley theory.

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We study the incompressible magneto-micropolar fluid equations with partial viscosity in R n n 2, 3 . A blow-up criterion of smooth solutions is obtained. The result is analogous to the celebrated Beale-Kato-Majda type criterion for the inviscid Euler equations of incompressible fluids.

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