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

Abstract

This paper concerns the lower bound decay rate of global solution for compressible Navier–Stokes–Korteweg system in three-dimensional whole space under the $$H^{4}\times H^{3}$$ 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 $$L^2$$ L 2 -norm is $$(1+t)^{-\frac{3}{4}}$$ ( 1 + t ) - 3 4 if the initial data satisfy some low-frequency assumption additionally. Furthermore, we also show that the lower bound of the $$k(k\in [1, 3])$$ k ( k ∈ [ 1 , 3 ] ) th-order spatial derivatives of solution converging to zero in $$L^2$$ L 2 -norm is $$(1+t)^{-\frac{3+2k}{4}}$$ ( 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 $$L^2$$ L 2 -norm is $$(1+t)^{-\frac{5}{4}}$$ ( 1 + t ) - 5 4 .

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

“…In other words, we will establish the claim estimates (), (), (), (), and () in the sequel. Even though the proof processes of several claim estimates below are similar to the claim estimates in Gao et al, 31,32 we still give proof in detail for completeness.…”

“…Notice that the first and second equation of the linearized part () are similar to the linearized system of compressible fluid models of Korteweg type in Gao et al 31 We still use the method in Gao et al 31 to obtain the lower bounds of decay rates of the solution to the system ()–(), which is stated as follows. For the sake of simplicity, we omit the proof here.…”

“…Now, we establish the upper and lower bounds of decay rates for the time derivative of solution ( ρ , u , B ) in the L 2 norm. The lower bounds of decay rates for the time derivative of density, velocity, and magnetic field can be obtained by using the method in Gao et al 31,32 Nevertheless, we still give the estimate in detail due to the appearance of the quantum potential and magnetic field simultaneously.…”