Space-time decay rate for the 3D compressible quantum magnetohydrodynamic model

This paper investigates the space-time decay properties of solutions to the three-dimensional compressible quantum magnetohydrodynamic (QMHD) model. By employing weighted Sobolev space techniques, we establish the optimal decay rate for the $k$-th order spatial derivatives of solutions with $k\in [0,4]$, which concides with the heat equation. Specifically, we prove that the decay rate in the weighted space $H_{\gamma }^2({\mathbb{R}}^3)$ is given by $t^{-\frac{3}{4}-\frac{k}{2}+\gamma }$ for the spatial derivatives of order $k$. The key contribution lies in developing a unified framework that connects the weighted energy estimates with time decay analysis, which enables us to simultaneously capture both the spatial regularity and temporal decay characteristics of the solution. This result generalizes the previous decay estimates and provides a new description of the solution's asymptotic behavior in quantum magnetohydrodynamic systems.