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

Bian, Dongfen; Yao, Lihua; Zhu, Changjiang

doi:10.1137/130942231

Cited by 39 publications

(24 citation statements)

References 25 publications

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“…Based on the above special choice for viscosities and capillary coefficients, they can take a linear combination of model (1.5) to reformulate it into two 4 × 4 systems whose linear parts are decoupled with each other and possess the same dissipation structure as that of the compressible Navier-Stokes-Korteweg system, and then employ the similar arguments as in [2,26] to prove their main results. However, since this reformulation played a crucial role in their analysis, the case of constant viscosities, even if the equal constant viscosities (i.e., µ ± (ρ ± ) = ν, λ ± (ρ ± ) = λ ), cannot be handled in their settings.…”

Section: Introductionmentioning

confidence: 99%

Global existence and decay rates for a generic compressible two-fluid model

Li,

Wang,

et al. 2021

Preprint

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We investigate global existence and optimal decay rates of a generic non-conservative compressible two-fluid model with general constant viscosities and capillary coefficients. Bresch, et al. in the seminal work (Arch Rational Mech Anal 196:599-629, 2010) considered the compressible two-fluid model with a special type of density-dependent viscosities (µ ± (ρ ± ) = µ ± ρ ± , λ ± = 0). However, as indicated by themselves, their methods cannot deal with the case of constant viscosity coefficients. Besides, Cui, et al. (SIAM J Math Anal 48:470-512, 2016) studied the same model with a more special type of density-dependent viscosity (µ ± (ρ ± ) = νρ ± , λ ± = 0) and equal capillary coefficients (σ + = σ − = σ ). Since their analysis relies heavily this special choice for viscosities and capillary coefficients, the case of general constant viscosities and capillary coefficients cannot be handled in their settings. The main novelty of this work is three-fold: First, for any integer ℓ ≥ 3, we show that the densities and velocities converge to their corresponding equilibrium states at the L 2 rate (1 + t) − 3 4 , and the k(∈ [1,ℓ])-order spatial derivatives of them converge to zero at the L 2 rate (1+t) − 3 4 − k 2 , which are the same as ones of the compressible Navier-Stokes system, Navier-Stokes-Korteweg system and heat equation. Second, the linear combination of the fraction densities (β + α + ρ + + β − α − ρ − ) converges to its corresponding equilibrium state at the L 2 rate (1 + t) − 3 4 , and its k(∈ [1,ℓ])-order spatial derivative converges to zero at the L 2 rate (1 + t) − 3 4 − k 2 , but the fraction densities (α ± ρ ± ) themselves converge to their corresponding equilibrium states at the L 2 rate (1 + t) − 1 4 , and the k(∈ [1,ℓ])-order spatial derivatives of them converge to zero at the L 2 rate (1 + t) − 1 4 − k 2 , which are slower than ones of their linear combination (β + α + ρ + + β − α − ρ − ) and the densities. We think that this phenomenon should owe to the special structure of the system. Finally, for well-chosen initial data, we also prove the lower bounds on the decay rates, which are the same as those of the upper decay rates. Therefore, these decay rates are optimal for the compressible two-fluid model.

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Section: Introductionmentioning

confidence: 99%

Global existence and decay rates for a generic compressible two-fluid model

Li,

Wang,

et al. 2021

Preprint

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“…The formulation of the theory of capillary with diffusive interface can be traced back to Korteweg, and a modern version was proposed by Dunn and Serrin . Since the compressible Navier‐Stokes‐Korteweg equations are the capillarity approximation of the classical compressible Navier‐Stokes equations (see Bian et al), one of the important topics about the compressible Navier‐Stokes‐Korteweg equations is to study the stability of basic waves such as viscous shock wave and rarefaction wave. More precisely, Chen and Li and Luo discussed asymptotic stability of the rarefaction waves for the one‐dimensional compressible Naviver‐Stokes‐Korteweg equation, respectively.…”

Section: Introductionmentioning

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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“…Charve and Haspot proved existence of the global strong solution of the 1‐dimensional compressible Navier‐Stokes‐Korteweg equations and then showed that the global strong solution converges to a weak‐entropy solution of the compressible Euler equations. Bian et al studied the capillarity limit of the 3‐dimensional compressible fluid models of Korteweg type to the Navier‐Stokes equations. Jüngel et al studied combined incompressible and vanishing capillarity limit in the barotropic compressible Navier‐Stokes equations for smooth solutions.…”

Section: Introductionmentioning

confidence: 99%

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

Peng

Wang

2018

Math Methods in App Sciences

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In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.

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

Cited by 39 publications

References 25 publications

Global existence and decay rates for a generic compressible two-fluid model

Global existence and decay rates for a generic compressible two-fluid model

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

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

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