Lower bound of decay rate for higher-order derivatives of solution to the compressible fluid models of Korteweg type

This paper concerns the lower bound decay rate of global solution for compressible Navier–Stokes–Korteweg system in three-dimensional whole space under the H 4 × H 3 framework. At first, the lower bound of decay rate for the global solution converging to constant equilibrium state (1, 0) in L 2 -norm is ( 1 + t ) - 3 4 if the initial data satisfy some low-frequency assumption additionally. Furthermore, we also show that the lower bound of the k ( k ∈ [ 1 , 3 ] ) th-order spatial derivatives of solution converging to zero in L 2 -norm is ( 1 + t ) - 3 + 2 k 4 . Finally, it is proved 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 ) - 5 4 .