Space-time decay rates of a two-phase flow model with magnetic field in R^3

We investigate the space-time decay rates of strong solution to a two-phase flow model with magnetic field in the whole space \(\mathbb{R}^3 \). Based on the temporal decay results by Xiao [24] we show that for any integer \(\ell\geq 3\), the space-time decay rate of \(k(0\leq k \leq \ell)\)-order spatial derivative of the strong solution in the weighted Lebesgue space \( L_\gamma^2 \) is \(t^{-\frac{3}{4}-\frac{k}{2}+\gamma}\). Moreover, we prove that the space-time decay rate of \(k(0\leq k \leq \ell-2)\)-order spatial derivative of the difference between two velocities of the fluid in the weighted Lebesgue space \( L_\gamma^2 \) is \(t^{-\frac{5}{4}-\frac{k}{2}+\gamma}\), which is faster than ones of the two velocities themselves.