On the Exponential Decay for Compressible Navier–Stokes–Korteweg Equations with a Drag Term

Bresch, Didier; Gisclon, Marguerite; Lacroix–Violet, Ingrid; Vasseur, Alexis

doi:10.1007/s00021-021-00639-2

Cited by 5 publications

(6 citation statements)

References 27 publications

(27 reference statements)

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“…We will base our proof of existence on this approach, with the specificity of taking an isothermal pressure law λρ (which leads to the use of energy with no definite sign, see below) and adding a dissipation term µρu. Note that in the recent paper [7] the authors have shown the exponential decay to equilibrium of global weak solutions of the Navier-Stokes-Korteweg system with such a dissipation term and for both barotropic and isothermal pressure laws.…”

Section: Introductionmentioning

confidence: 97%

Global dissipative solutions of the defocusing isothermal Euler-Langevin-Korteweg equation

Chauleur¹

2020

Preprint

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We construct global dissipative solutions on the torus of dimension at most three of the defocusing isothermal Euler-Langevin-Korteweg system, which corresponds to the Euler-Korteweg system of compressible quantum fluids with an isothermal pressure law and a linear drag term with respect to the velocity. In particular, the isothermal feature prevents the energy and the BD-entropy from being positive. Adapting standard approximation arguments we first show the existence of global weak solutions to the defocusing isothermal Navier-Stokes-Langevin-Korteweg system. Introducing a relative entropy function satisfying a Gronwall-type inequality we then perform the inviscid limit to obtain the existence of dissipative solutions of the Euler-Langevin-Korteweg system. Contents 1. Introduction 1 2. The isothermal Navier-Stokes-Langevin-Korteweg system 5 2.1. Regularized NSLK system 8 2.2. NSLK with drag forces 12 2.3. The limit r 0 , r 1 → 0 13 3. The isothermal Euler-Langevin-Korteweg system 15 3.1. A Gronwall inequality 17 3.2. The viscous limit ν → 0 21 Appendix A. Definition of the operators and technical lemmas 23 References 24

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Section: Introductionmentioning

confidence: 97%

Global dissipative solutions of the defocusing isothermal Euler-Langevin-Korteweg equation

Chauleur¹

2020

Preprint

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“…In order to obtain the large time behavior (decay exponentially) of weak solutions, we obtain the relative entropy inequality between weak solutions and equilibrium solutions by introducing the relative entropy functional. Considering that the structure of reduced gravity two-and-a-half layer model is more complicated than compressible Navier-Stokes equations due to the presence of cross terms h 1 ∇h 2 , h 2 ∇h 1 , we need to estimate the cross term g 2 Ω (h 1 − R 1 )(h 2 − R 2 ) dx in relative entropy, which is different from the work of Bresch, Gisclon, Lacroix-Violet, Vasseur [7]. It should point out that g 2 Ω (h 1 − R 1 )(h 2 − R 2 ) dx can not be directly controlled by the cross term Ω ∇h 1 • ∇h 2 dx.…”

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confidence: 99%

“…Note that the method of Su-Li-Yao [26] can be applied to prove global existence of weak solution to the problem (1.1)-(1.2). The work of this paper is inspired by Bresch, Gisclon, Lacroix-Violet [6] and Bresch, Gisclon, Lacroix-Violet, Vasseur [7]. In order to obtain the large time behavior (decay exponentially) of weak solutions, we obtain the relative entropy inequality between weak solutions and equilibrium solutions by introducing the relative entropy functional.…”

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confidence: 99%

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Exponential decay for 2D reduced gravity two-and-a-half layer model with quantum potential and drag force

Yao

Zhu

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we study the global weak solutions to a reduced gravity two-and-a-half layer model with quantum potential and drag force in two-dimensional torus. Inspired by Bresch, Gisclon, Lacroix-Violet [Arch. Ration. Mech. Anal. (233):975-1025, 2019] and Bresch, Gisclon, Lacroix-Violet, Vasseur [J. Math. Fluid Mech., 24(11):16, 2022], we prove that the weak solutions decay exponentially in time to equilibrium state. In order to prove the decay property of weak solutions, we obtain the relative entropy inequality of weak solutions and equilibrium solutions by defining the relative entropy functional. Considering that the structure of reduced gravity two-and-a-half layer model is more complicated than the compressible Navier-Stokes equations due to the presence of cross terms <inline-formula><tex-math id="M1">\begin{document}$ h_{1}\nabla h_{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ h_{2}\nabla h_{1} $\end{document}</tex-math></inline-formula>, we need to estimate the cross term in relative entropy.</p>

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Effect of Temperature Upon Double Diffusive Instability in Navier–Stokes–Voigt Models with Kazhikhov–Smagulov and Korteweg Terms

Straughan

2023

Appl Math Optim

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We present models for convection in a mixture of viscous fluids when the layer is heated from below and simultaneously the pointwise volume concentration of one of the fluids is heavier below. This configuration produces a problem of competitive double diffusion since heating from below promotes instability, but the greater density of fluid below is stabilizing. The fluids are of linear viscous type which may contain Kelvin–Voigt terms, but density gradients due to the mixture appear strongly in the governing equations. The density gradients give rise to Korteweg stresses, but may also be described by theory due to Kazhikhov and Smagulov. The systems of equations which appear are thus highly nonlinear. The instability surface threshold is calculated and this is found to have a complex nonlinear shape, very different from the linear ones found in classical thermohaline convection in a Navier–Stokes fluid. It is shown that the Kazhikhov–Smagulov terms, Korteweg terms and Kelvin–Voigt term play a key role in acting as stabilizing agents but the associated effect is very nonlinear. Quantitative values of the instability surface are displayed showing the effect Korteweg terms, Kazhikhov–Smagulov terms, and the Kelvin Voigt term have. The nonlinear stability problem is addressed by means of a generalized energy theory deriving different results depending on which underlying theory is employed.

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On the Exponential Decay for Compressible Navier–Stokes–Korteweg Equations with a Drag Term

Cited by 5 publications

References 27 publications

Global dissipative solutions of the defocusing isothermal Euler-Langevin-Korteweg equation

Global dissipative solutions of the defocusing isothermal Euler-Langevin-Korteweg equation

Exponential decay for 2D reduced gravity two-and-a-half layer model with quantum potential and drag force

Effect of Temperature Upon Double Diffusive Instability in Navier–Stokes–Voigt Models with Kazhikhov–Smagulov and Korteweg Terms

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