Lower bound of decay rate for higher-order derivatives of solution to the compressible fluid models of Korteweg type
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Lower bound of decay rate for higher-order derivatives of solution to the compressible fluid models of Korteweg type Jincheng Gao, Zeyu Lyu
and Zheng-an Yao
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 × H 3 framework. At first, the lower bound of decay rate for the 3 global solution converging to constant equilibrium state (1, 0) in L2 -norm is (1 + t)− 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 L2 -norm is (1 + t)−
3+2k 4
. Finally, it is proved that the lower bound of decay 5
rate for the time derivatives of density and velocity converging to zero in L2 -norm is (1 + t)− 4 . Mathematics Subject Classification. 35Q35, 35B40. Keywords. Compressible Navier–Stokes–Korteweg system, Lower bound decay.
1. Introduction In this paper, 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, (1.1) (ρu)t + div(ρu ⊗ u) − μΔu − (μ + ν)∇divu + ∇P (ρ) = κρ∇Δρ, where t ≥ 0 is time, x ∈ R3 is spatial coordinate and the unknown functions ρ = ρ(x, t) and u = (u1 , u2 , u3 )(x, t) represent density and velocity, respectively. The pressure P (ρ) is a smooth function in a neighborhood of 1 with P (1) > 0. The constant viscosity coefficients μ and ν satisfy the following physical conditions: μ > 0, 2μ + 3ν ≥ 0. The constant capillary coefficient κ satisfies κ > 0. To complete the system (1.1), the initial data are given by (ρ, u)(x, t)|t=0 = (ρ0 (x), u0 (x)).
(1.2)
Furthermore, as the space variable tends to infinity, we assume lim (ρ0 − 1, u0 )(x) = 0.
|x|→∞
(1.3)
Korteweg-type models, supposing that the energy of the fluid depends on standard variables and the gradient of the density, are based on an extended version of nonequilibrium thermodynamics. The Navier– Stokes–Korteweg model, describing the dynamics of a liquid–vapor mixture with diffuse interphase and being used to model the motion of compressible fluid with capillary effect of materiel, originals from the work of van der Waals [23] and Korteweg [17], and the modern form was derived by Dunn and Serrin [8]. It should be noted that the system (1.1) will reduce to the well-known compressible Navier–Stokes system if the capillary coefficient satisfies κ = 0. 0123456789().: V,-vol
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J. Gao, Z. Lyu and Z. Yao
ZAMP
There are many studies on the well-posedness of solutions to the compressible fluid models of Korteweg type. For the one-dimensional case, many researchers have studied extensively; refer to [3,5] and the references therein. Charve and Haspot [3] obtained the global strong solution in the case of Saint-Venant viscosity coefficients. Chen et al. [5] obtained the global existence of classical sol
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