Lower bound of decay rate for higher order derivatives of solution to the compressible quantum magnetohydrodynamic model

The lower bounds of decay rates for global solution to the compressible viscous quantum magnetohydrodynamic model in three‐dimensional whole space under the H 5  ×  H 4  ×  H 4 framework are investigated in this paper. We first show that the lower bound of decay rate for the solution converging to constant equilibrium state (1, 0, 0) in L 2 ‐norm is( 1 + t ) − 3 4when the initial data satisfy some low‐frequency assumption. Moreover, we prove that the lower bound of decay rate of k ( k  ∈ [1, 3]) order spatial derivative for the solution converging to constant equilibrium state (1, 0, 0) in L 2 ‐norm is( 1 + t ) − 3 + 2 k 4. Then, we show that the lower bound of decay rate for the time derivatives of density and velocity is( 1 + t ) − 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 ) − 7 4.