Stability properties of the Euler–Korteweg system with nonmonotone pressures

Giesselmann, Jan; Tzavaras, Athanasios E.

doi:10.1080/00036811.2016.1276175

Cited by 27 publications

(15 citation statements)

References 19 publications

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“…Nonlocal Cahn-Hilliard problems have been introduced in [12] as models for phase separation dynamics. Another asymptotic regime for Korteweg fluids that is governed by the Cahn-Hilliard equation can be found in [13].…”

Section: The Main Resultsmentioning

confidence: 99%

Homogenization of Non-Local Navier-Stokes-Korteweg Equations for Compressible Liquid-Vapour Flow in Porous Media

Rohde¹,

Wolff²

2019

Preprint

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We consider a nonlocal version of the quasi-static Navier-Stokes-Korteweg equations with a non-monotone pressure law. This system governs the low-Reynolds number dynamics of a compressible viscous fluid that may take either a liquid or a vapour state. For a porous domain that is perforated by cavities with diameter proportional to their mutual distance the homogenization limit is analyzed. We extend the results for compressible one-phase flow with polytropic pressure laws and prove that the effective motion is governed by a nonlocal version of the Cahn-Hilliard equation. Crucial for the analysis is the convolution-like structure of the nonlocal capillarity term that allows to equip the system with a generalized convex free energy. Moreover, the capillarity term accounts not only for the energetic interaction within the fluid but also for the interaction with a solid wall boundary.

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Section: The Main Resultsmentioning

confidence: 99%

Homogenization of Non-Local Navier-Stokes-Korteweg Equations for Compressible Liquid-Vapour Flow in Porous Media

Rohde¹,

Wolff²

2019

Preprint

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“…The multi‐dimensional equation often appears in the context of semi‐conductor modeling, and can be viewed as the evolution of the density of electrons with vanishing temperature of the simplified quantum drift‐diffusion model [1, 12]:

\partial_{t} u = div false(T \nabla u + u \nabla V false), V = V_{e} - \frac{ϵ^{2}}{6} \frac{Δ \sqrt{u}}{\sqrt{u}} .

Here

T > 0

is the temperature,

ϵ

the Planck constant, and

V

the potential felt by the electrons, which splits into the classical electric potential

V_{e}

and the Bohm potential, describing quantum effects. The equation can also be found in quantum hydrodynamics as the high friction limit of some quantum hydrodynamic equations, see References [14, 15, 21].…”

Section: Introductionmentioning

confidence: 99%

A positivity‐preserving and energy stable scheme for a quantum diffusion equation

Huo

Liu

2021

Numerical Methods Partial

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We propose a new fully‐discretized finite difference scheme for a quantum diffusion equation, in both one and two dimensions. This is the first fully‐discretized scheme with proven positivity‐preserving and energy stable properties using only standard finite difference discretization. The difficulty in proving the positivity‐preserving property lies in the lack of a maximum principle for fourth order partial differential equations. To overcome this difficulty, we reformulate the scheme as an optimization problem based on a variational structure and use the singular nature of the energy functional near the boundary values to exclude the possibility of non‐positive solutions. The scheme is also shown to be mass conservative and consistent.

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“…Work of Sheffer [25] has shown that there is no hope for such a trait even for the distributional solutions of Euler's equations, therefore the main point of [7] is to introduce some kind of energy conservation property from which the single solution would emerge. This "relative energy method" or "relative entropy method" was first used by Dafermos ([11], [12]) in regard to the scalar conservation laws and it is used to describe how two physical systems differ in time, starting from close initial states; its other applications range from stability studies, asymptotic limits (see [10], [18], [17], [6]) to problems arising from biology ( [8], [13], [20], [16]).…”

Section: Introductionmentioning

confidence: 99%

Mv-strong uniqueness for density dependent, non-Newtonian, incompressible fluids

Woźnicki

2021

Preprint

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We consider density dependent, non-Newtonian, incompressible system with the space being flat torus. The viscious stress in the momentum equation is understood through the rheological law and its connection to the proper convex potential. We define the dissipative measure-valued solutions for the aforementioned equations as well as provide a proof of its existence. The main result of this work is the mv-strong uniqueness of the defined solutions.

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Stability properties of the Euler–Korteweg system with nonmonotone pressures

Cited by 27 publications

References 19 publications

Homogenization of Non-Local Navier-Stokes-Korteweg Equations for Compressible Liquid-Vapour Flow in Porous Media

Homogenization of Non-Local Navier-Stokes-Korteweg Equations for Compressible Liquid-Vapour Flow in Porous Media

A positivity‐preserving and energy stable scheme for a quantum diffusion equation

Mv-strong uniqueness for density dependent, non-Newtonian, incompressible fluids

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