Large Time Behavior of Global Strong Solutions to a Two-Phase Model with a Magnetic Field

In this paper, the Cauchy problem for a two-phase model with a magnetic field in three dimensions is considered. Based on a new linearized system with respect to (c − c, P − P, u, H) for constants c ≥ 0 and P > 0, the existence theory of global strong solution is established when the initial data is close to its equilibrium in three dimensions for the small H initial data. We improve the existence results obtained by Wen and Zhu in [40] where an additional assumption that the initial perturbations are bounded in L-norm was needed. The energy method combined with the low-frequency and high-frequency decomposition is used to derive the decay of the solution and hence the global existence. As a by-product, the time decay estimates of the solution and its derivatives in the L-norm are obtained.