*Mathematical Methods in the Applied Sciences*Lower Bound of Decay Rate for Higher Order Derivatives of Solution to
the Compressible Quantum Magnetohydrodynamic Model

The lower bound decay rate of global solution to the compressible
viscous quantum magnetohydrodynamic model in three-dimensional whole
space under the $H^5\times
H^4\times H^4$ framework is investigated in this
paper. We firstly show that the lower bound of decay rate for the
density, velocity and magnetic field converging to the equilibrium state
(1,0,0) in $L^2$-norm is
$(1+t)^{-\frac{3}{4}}$ when the initial data
satisfies some low frequency assumption. Moreover, we prove that the
lower bound of decay rate of $k(k\in [1,3])$ order
spatial derivative for the density, velocity and magnetic field
converging to the equilibrium state (1,0,0) in $L^2$-norm is
$(1+t)^{-\frac{3+2k}{4}}$. Then we show that
the lower bound of decay rate for the time derivatives of density and
velocity converging to zero in $L^2$-norm is
$(1+t)^{-\frac{5}{4}}$, but the lower bound of
decay rate for the time derivative of magnetic field converging to zero
in $L^2$-norm is $(1+t)^{-\frac{7}{4}}$.

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