In this paper, we consider the compressible fluid model of Korteweg type which can be used as a phase transition model. It is shown that the system admits a unique, global strong solution for small...

The Global Well-Posedness for the Compressible Fluid Model of Korteweg Type

This paper is concerned with the time-asymptotic behavior of strong solutions to the initial-boundary value problem of the isothermal compressible fluid models of Korteweg type with density-depende...

Asymptotic Behavior of Solutions to An Impermeable Wall Problem of the Compressible Fluid Models of Korteweg Type with Density-dependent Viscosity and Capillarity

This study is concerned with the large time behaviour of the three-dimensional isentropic compressible Navier–Stokes–Korteweg equations, which are used to model viscous and compressible fluids with internal capillarity. Based on the fact that the rarefaction wave is nonlinearly stable to the one-dimensional isentropic compressible Navier–Stokes–Korteweg equations, the planar rarefaction wave to the three-dimensional isentropic compressible Navier–Stokes–Korteweg equations is first constructed. Then it is shown that the planar rarefaction wave is asymptotically stable in the case that the initial data are a suitably small perturbation of the planar rarefaction wave and the strength of the rarefaction wave is small. The proof is based on the delicate energy method. The result indicate that the planar rarefaction wave of the inviscid Euler system is stable for the three-dimensional isentropic compressible fluids with physical viscosities and internal capillarity.

Stability of the planar rarefaction wave to three-dimensional Navier–Stokes–Korteweg equations of compressible fluids

Existence of smooth solutions for the compressible barotropic Navier‐Stokes‐Korteweg system without increasing pressure law

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 $$k(k\in [1, 3])$$
 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 $$(1+t)^{-\frac{5}{4}}$$
 .

/pdf/lower-bound-of-decay-rate-for-higher-order-derivatives-of-1mk7y58hhi.pdf

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

In this paper, we investigate the global well-posedness of classical solutions to three-dimensional Cauchy problem of the compressible Navier–Stokes type system with a Korteweg stess tensor under the condition that the initial energy is small. This result improves previous results obtained by Hattori–Li in [H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal. 25 (1994) 85–98; H. Hattori and D. Li. Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198 (1996) 84–97.], where the existence of the classical solution is established for initial data close to an equilibrium in some Sobolev space Hs.

Global classical solutions to a 3D Navier–Stokes–Korteweg equations with small initial energy

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Global classical solutions to a 3D Navier–Stokes–Korteweg equations with small initial energy