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 .

The lower bounds of decay rates for global solution to the compressible viscous quantum magnetohydrodynamic model in three‐dimensional whole space under the H5 × H4 × H4 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 L2‐norm is false(1+tfalse)−34 when 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 L2‐norm is false(1+tfalse)−3+2k4. Then, we show that the lower bound of decay rate for the time derivatives of density and velocity is false(1+tfalse)−54, but the lower bound of decay rate for the time derivative of magnetic field converging to zero in L2‐norm is false(1+tfalse)−74.

This a preprint and has not been peer reviewed. Data may be preliminary.

Recently, Bjorland and Schonbek [Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 907-936] investigated the upper bound of the decay rate for the solution to the Lagrangian-averaged Navier-Stokes-˛equation under the condition that the initial data belongs to L 1 .R n / \ H N .R n / with n D 2; 3; 4. The decay rate can eventually be shown to be optimal if the average of the initial data is nonzero. Thus, the target in this paper is to study the optimal decay rate of the solution when the average of the initial data is zero. If the initial data belongs to L 1 .R 3 / \ H N .R 3 / and some weighted Sobolev space, we show that the lower and upper bounds of decay rates for the kthorder (k 2 OE0; N ) spatial derivatives of the solution tending to zero in L 2 -norm are .1 C t/ 5C2k 4 , which implies these decay rates are optimal. As a by-product, we show that the optimal decay rate (including lower and upper bounds) of the time derivative of the solution tending to zero in L 2 -norm is .1 C t / 9 4 .