Space-time decay rate for strong solutions to the viscous liquid-gas two-phase flow with magnetic field

We investigate the viscous liquid-gas two-phase model with magnetic field in a weighted Sobolev space $H^{2}_{ a }(\mathbb {R}^{3})$. Based on precise weighted energy estimation, we establish the space-time decay rates of the $k(\in [0, 2])$th order spatial derivative of strong solutions. The main difficulty comes from the lack of dissipative structure of $\| \nabla P \|_{L^{2}_{a}}^{2}$, and we need to construct an interactive weighted energy functional to solve it.