Solutions for Two-Dimensional System for Materials of Korteweg Type

Hattori, Harumi; Li, Dening

doi:10.1137/s003614109223413x

Cited by 123 publications

(86 citation statements)

References 8 publications

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“…It should be noted that the uniform lower and upper bounds of the density function ρ ( t , x , y ) in guarantee the strict parabolicity of the momentum equation, which are crucial for the local and global‐in‐time existence of the classical solution to the system . In Hattori and Li, the authors have showed the local existence theorem, so we omit the details here. On the other hand, the a priori estimates have been given in Proposition .…”

Section: The Proof Of Theoremmentioning

confidence: 99%

Stability of the planar rarefaction wave to two‐dimensional Navier‐Stokes‐Korteweg equations of compressible fluids

Chen

Luo

2019

Math Methods in App Sciences

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This study is concerned with the large time behavior of the two‐dimensional compressible Navier‐Stokes‐Korteweg equations, which are used to model compressible fluids with internal capillarity. Based on the fact that the rarefaction wave, one of the basic wave patterns to the hyperbolic conservation laws is nonlinearly stable to the one‐dimensional compressible Navier‐Stokes‐Korteweg equations, the planar rarefaction wave to the two‐dimensional compressible Navier‐Stokes‐Korteweg equations is first derived. Then, it is shown that the planar rarefaction wave is asymptotically stable in the case that the initial data are suitably small perturbations of the planar rarefaction wave. The proof is based on the delicate energy method. This is the first stability result of the planar rarefaction wave to the multi‐dimensional viscous fluids with internal capillarity.

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Section: The Proof Of Theoremmentioning

confidence: 99%

Stability of the planar rarefaction wave to two‐dimensional Navier‐Stokes‐Korteweg equations of compressible fluids

Chen

Luo

2019

Math Methods in App Sciences

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“…The proof of Proposition 2.1 is standard, which is similar to that of Theorem 1.1 in [6] and thus omitted here for brevity. To prove Theorem 2.1, it remains to show the following a priori estimates by the standard continuation argument.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

“…There have been many mathematical results on the compressible fluid models of Korteweg type. In two or more space dimensions, Hattori and Li proved the local existence and global existence of smooth solutions for the isothermal [6,7] and nonisothermal [8] compressible fluid models of Korteweg type in Sobolev space. Bresch, Desjardins, and Lin [1] showed the global existence of weak solutions for an isothermal fluid in a periodic box or strip domain and such a result was later improved by Haspot in [9].…”

Section: Introductionmentioning

confidence: 99%

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

Chen

Chai

Dong

et al. 2015

Journal of Differential Equations

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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) = ρ α and the capillarity coefficient κ(ρ) = ρ β 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].

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“…An important question is the solvability of the isothermal NSK equations, which has received considerable attention. For isothermal NSK equations, local and global smooth solutions for Cauchy problems of (1) with constant coefficients and small, smooth initial data were discussed in [21,22]; the extension to Lipschitz continuous viscous coefficients and more general initial conditions was presented in [26]. In [33] a mathematical model with physically relevant non-local energies was proposed instead of the Van der Waals free energy and a short-time existence theorem for the Cauchy problem of the non-local NSK equations was proved.…”

Section: Introductionmentioning

confidence: 99%

A local discontinuous Galerkin method for the (non)-isothermal Navier–Stokes–Korteweg equations

Tian

Kuerten

et al. 2015

Journal of Computational Physics

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In this article, we develop a local discontinuous Galerkin (LDG) discretization of the (non)-isothermal Navier-Stokes-Korteweg (NSK) equations in conservative form. These equations are used to model the dynamics of a compressible fluid exhibiting liquid-vapour phase transitions. T he NSK-equations are closed with a Van der Waals equation of state and contain third order nonlinear derivative terms. These contributions frequently cause standard numerical methods to violate the energy dissipation relation and require additional stabilization terms to prevent numerical instabilities. In order to address these problems we first develop an LDG method for the isothermal NSK equations using discontinuous finite element spaces combined with a time-implicit Runge-Kutta integration method. Next, we extend the LDG discretization to the non-isothermal NSK equations. An important feature of the LDG discretizations presented in this article is that they are relatively simple, robust and do not require special regularization terms. Finally, computational experiments are provided to demonstrate the capabilities, accuracy and stability of the LDG discretizations.

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Solutions for Two-Dimensional System for Materials of Korteweg Type

Cited by 123 publications

References 8 publications

Stability of the planar rarefaction wave to two‐dimensional Navier‐Stokes‐Korteweg equations of compressible fluids

Stability of the planar rarefaction wave to two‐dimensional Navier‐Stokes‐Korteweg equations of compressible fluids

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

A local discontinuous Galerkin method for the (non)-isothermal Navier–Stokes–Korteweg equations

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