Optimal decay of the magnetohydrodynamic model for quantum plasmas with potential force

This paper considers the quantum magnetohydrodynamic model for quantum plasmas with potential force. We prove the optimal decay rates for the higher order derivative of the global small solution to the stationary state in the whole space. Specially, we show the optimal $ \dot{H}^{k} $ decay rate $ \sim (1+t)^{\frac{3}{4}+\frac{k}{2}} $, which improves the work of Xu and Pu (Discrete Contin. Dyn. Syst. Ser. B, 26, 2021). The proof is based on the optimal decay of the linearized equations, multi-frequency decompositions, and nonlinear energy estimates.