Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 41, pp. 1-29.

Title: Space-time decay rates of a two-phase flow model with magnetic field in R^3
       
Authors: Qin Ye (Guangxi Normal Univ., Guilin, Guangxi, China)
         Yinghui Zhang (Guangxi Normal Univ., Guilin, Guangxi, China)

Abstract: 
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. 

Submitted September 21, 2022. Published June 23, 2023.
Math Subject Classifications: 35Q31, 35K65, 76N10.
Key Words: Compressible Euler equations; Two-phase flow model; Space-time decay rate; Weighted Sobolev space.