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} 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)^{-frac{3}{4}} if the initial data satisfy some low-frequency assumption additionally. Furthermore, we also show that the lower bound of the k(kin [1, 3])th-order spatial derivatives of solution converging to zero in L^2-norm is (1+t)^{-frac{3+2k}{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)^{-frac{5}{4}}.
Highlights
We are concerned with the lower bounds of decay rate for the global solution to the compressible Navier–Stokes–Korteweg system in three-dimensional whole space: ρt + div(ρu) = 0,t + div(ρu ⊗ u) − μΔu − (μ + ν)∇divu + ∇P (ρ) = κρ∇Δρ, (1.1)
The lower bounds of decay rates (1.9), (1.10) for the derivatives of density and velocity to the compressible Navier–Stokes–Korteweg system are obtained for the first time
We give the lower bound of decay rate for the higher-order spatial derivative of solution to the compressible Navier–Stokes–Korteweg equations
Summary
Chen et al [5] obtained the global existence of classical solutions with large initial data away from vacuum for the isothermal compressible fluid of Korteweg type under the condition that the viscosity coefficient and capillarity coefficient are dependent on the density. The lower bounds of decay rates (1.9), (1.10) for the derivatives of density and velocity to the compressible Navier–Stokes–Korteweg system are obtained for the first time. If we use the transport equation to obtain the lower bound of decay rate for the time derivative of density, we need to get the lower bound for the quantity divergence of velocity(i.e., divu) To achieve this target, we need to assume the smallness for the initial velocity in L1.
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