Fluid Models Of Korteweg Type Research Articles

Vanishing capillarity limit of the compressible non-isentropic Navier–Stokes–Korteweg system to Navier–Stokes system

In this paper, we consider the three dimensional compressible nonisentropic fluid model of Korteweg type. We mainly present the vanishing capillarity limit of smooth solution to the initial value problem. Based on the uniform estimates with respect to the capillary coefficient κ, we show the unique smooth solutions of the three dimensional Korteweg model converges globally in time to the smooth solution of the three dimensional compressible nonisentropic Navier–Stokes system, as κ tends to zero. Also, we give the convergence rate estimates in H2-norm with respect to κ.

Large-time behavior of smooth solutions to the isothermal compressible fluid models of Korteweg type with large initial data

This paper is concerned with the large-time behavior of smooth non-vacuum solutions with large initial data to the Cauchy problem of the one-dimensional isothermal compressible fluid models of Korteweg type with the viscosity coefficient μ(ρ)=ρα and the capillarity coefficient κ(ρ)=ρβ. Here α∈R and β∈R are some parameters. Depending on whether the far-fields of the initial data are the same or not, we prove that the corresponding Cauchy problem admits a unique global smooth solution which tends to constant states or rarefaction waves respectively, as time goes to infinity, provided that α and β satisfy some conditions. Note that the initial perturbation can be arbitrarily large. The proofs are given by the elementary energy method and Kanel’s technique (Kanel, 1968). Compared with former results in this direction obtained by Germain and LeFloch (2016), and Chen et al. (2015), the main novelties of this paper lie in the following: First, we obtain the global existence of smooth solutions with large data for some new varieties of parameters α and β. Second, the large-time behavior of smooth large solutions around constant states is established.

Global classical solutions to the one-dimensional compressible fluid models of Korteweg type with large initial data

This paper is concerned with the global existence of classical solutions with large initial data away from vacuum to the Cauchy problem of the one-dimensional isothermal compressible fluid models of Korteweg type with density-dependent viscosity coefficient and capillarity coefficient. The case when the viscosity coefficient μ(ρ)=ρα and the capillarity coefficient κ(ρ)=ρβ for some parameters α,β∈R is considered. Under some conditions on α,β, we first show the global existence of large solutions around constant states if the far-fields of the initial data are the same, while if the far-fields of the initial data are different, we prove the global stability of rarefaction waves with large strength. Here global stability means the initial perturbation can be arbitrarily large. Our analysis is based on the elementary energy method and the technique developed by Y. Kanel' [29].

Vanishing capillarity–viscosity limit for the incompressible inhomogeneous fluid models of Korteweg type

In this paper, we investigate the convergence rates of the solution to the incompressible inhomogeneous fluid models of Korteweg type in three-dimensional whole space. We prove that as the capillarity coefficient and viscosity coefficient vanish, the local-in-time solution of the Cauchy problem for this system converges to the solution of the incompressible inhomogeneous Euler equations. In the end, we obtain a continuation theorem for the local smooth solution.

Nonlinear stability of traveling wave solutions for the compressible fluid models of Korteweg type

This paper is concerned with the existence and time-asymptotic nonlinear stability of traveling wave solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. The existence of traveling wave solutions is obtained by the phase plane analysis, then the traveling wave solution is shown to be asymptotically stable by the elementary L2-energy method.

Vanishing viscosity limit to rarefaction waves for the full compressible fluid models of Korteweg type

We prove the solution of the full compressible fluid models of Korteweg type with centered rarefaction wave data of large strength exists globally in time. As the viscosity, heat-conductivity and capillary coefficients tend to zero, the global solution converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly when the initial perturbation is small. Our analysis is based on the energy method.

Vanishing Capillarity Limit of the Compressible Fluid Models of Korteweg Type to the Navier--Stokes Equations

In this paper, we consider the three-dimensional isentropic compressible fluid models of Korteweg type, called the compressible Navier--Stokes--Korteweg system. We mainly present the vanishing capillarity limit of the smooth solution to the initial value problem. Precisely, we first establish the uniform estimates of the global smooth solution with respect to the capillary coefficient $\kappa$. Then by the Lions--Aubin lemma, we show that the unique smooth solution of the three-dimensional Navier--Stokes--Korteweg system converges globally in time to the smooth solution of the three-dimensional Navier--Stokes system as $\kappa$ tends to zero. Also, we give the convergence rate estimates for any given positive time.

Nonlinear stability of viscous contact wave for the one‐dimensional compressible fluid models of Korteweg type

This paper is concerned with the large time behavior of solutions of the Cauchy problem to the one‐dimensional compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. When the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, it is shown that the viscous contact wave corresponding to the contact discontinuity is asymptotically stable provided that the strength of contact discontinuity and the initial perturbation are suitably small. The analysis is based on the elementary L2‐energy method together with continuation argument. Copyright © 2013 John Wiley & Sons, Ltd.

Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type

This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimension compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. Assume that the corresponding Riemann problem to the compressible Euler system can be solved by rarefaction waves (VR,UR)(t,x). If the initial data is a small perturbation of an approximate rarefaction wave for (VR,UR)(t,x), we show that the corresponding Cauchy problem admits a unique global smooth solution which tends to (VR,UR)(t,x) time asymptotically. Since we do not require the strength of the rarefaction waves to be small, this result gives the nonlinear stability of strong rarefaction waves for the one-dimensional compressible fluid models of Korteweg type. The analysis is based on the elementary L2 energy method together with continuation argument.

Blow-up of smooth solutions to the compressible fluid models of Korteweg type

We show the blow-up of smooth solutions to a non-isothermal model of capillary compressible fluids in arbitrary space dimensions with initial density of compact support. This is an extension of Xin’s result [Xin, Z.: Blow-up of smooth solutions to the compressible Navier-Stokes equations with compact density. Comm. Pure Appl. Math., 51, 229–240 (1998)] to the capillary case but we do not need the condition that the entropy is bounded below. Moreover, from the proof of Theorem 1.2, we also obtain the exact relationship between the size of support of the initial density and the life span of the solutions. We also present a sufficient condition on the blow-up of smooth solutions to the compressible fluid models of Korteweg type when the initial density is positive but has a decay at infinity.

Large time behavior of solutions to the isentropic compressible fluid models of Korteweg type in R³

Large time behavior of solutions to the isentropic compressible fluid models of Korteweg type in R³

Global existence and optimal decay rate of the compressible Navier–Stokes–Korteweg equations with external force

In this paper, we consider the three-dimensional compressible fluid models of Korteweg type. The equation takes the form of the compressible Navier–Stokes equation, with the Cauchy stress tensor in the momentum equations. Under the smallness assumption of the external force in some Sobolev space, we first show the existence of stationary solution by standard iterative arguments. Next, we present that the global-in-time solution of the initial value problem for the three-dimensional compressible Navier–Stokes–Korteweg equations exists uniquely and approaches the stationary solution as t→∞, provided the prescribed initial data and the external force are sufficiently small. Finally, based on the elaborate energy estimates of the solution for the nonlinear system and L2-decay estimates of semigroup generated by the corresponding linearized equation, we show the optimal L2-convergence rates of the solution towards the stationary solution. As far as we know, this is the first result about the global existence and L2-decay rate of smooth solutions for the compressible Navier–Stokes–Korteweg equations with the external force.

Strong solutions for the incompressible fluid models of Korteweg type

In this article, we are concerned with the strong solutions for the incompressible fluid models of Korteweg type in a bounded domain Ω ⊂ ℝ 3 . We prove the existence and uniqueness of local strong solutions to the initial boundary value problem. We point out that in this article we allow the existence of initial vacuum provided initial data satisfy a compatibility condition.

Existence of solutions for compressible fluid models of Korteweg type

The purpose of this work is to prove existence and uniqueness results of suitably smooth solutions for an isothermal model of capillary compressible fluids derived by J.E. Dunn and J. Serrin (1985), which can be used as a phase transition model.We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence for solutions when the pressure law may present spinodal regions. Uniqueness is also obtained.Assuming a lower and upper control of the density, we also show the existence of weak solutions in dimension 2 near equilibrium. Finally, referring to the work of Z. Xin (1998) in the non-capillary case, we describe some blow-up properties of smooth solutions with finite total mass.