Regularity and separation from potential barriers for a non-local phase-field system

Londen, Stig–Olof; Petzeltová, Hana

doi:10.1016/j.jmaa.2011.02.003

Cited by 29 publications

(49 citation statements)

References 18 publications

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“…and the nonlocal chemical potential given by (1.4) is obtained by taking the first variation of E. The physical relevance of nonlocal interactions was already pointed out in the pioneering paper [57] (see also [24, 4.2] and references therein) and studied (in the case of constant velocity) for different kind of evolution equations, mainly Cahn-Hilliard and phase field systems, see, e.g., [10,19,35,36,37,38,53,54,48,49,34]. Diffuse interface models for two-phase flow of fluids with identical densities are very well established and studied in literature.…”

Section: Introductionmentioning

confidence: 99%

Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

Frigeri

2016

Math. Models Methods Appl. Sci.

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Abstract. We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists in a Navier-Stokes type system coupled with a convective nonlocal Cahn-Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

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Section: Introductionmentioning

confidence: 99%

Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

Frigeri

2016

Math. Models Methods Appl. Sci.

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“…Also, the standard CH equation can be interpreted as an approximation of the nonlocal one. The nonlocal CH equation has been analyzed in a number of papers, under various assumptions on the potential F and on the mobility (see, e.g., [1,7,27,17,18,24,25,29], cf. also [22,23] for the numerics).…”

Section: Introductionmentioning

confidence: 99%

On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems

Porta¹,

Grasselli²

2016

CPAA

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The phase separation of an isothermal incompressible binary fluid in a porous medium can be described by the so-called Brinkman equation coupled with a convective Cahn-Hilliard (CH) equation. The former governs the average fluid velocity u, while the latter rules evolution of ϕ, the difference of the (relative) concentrations of the two phases. The two equations are known as the Cahn-Hilliard-Brinkman (CHB) system. In particular, the Brinkman equation is a Stokes-like equation with a forcing term (Korteweg force) which is proportional to µ∇ϕ, where µ is the chemical potential. When the viscosity vanishes, then the system becomes the Cahn-Hilliard-Hele-Shaw (CHHS) system. Both systems have been studied from the theoretical and the numerical viewpoints. However, theoretical results on the CHHS system are still rather incomplete. For instance, uniqueness of weak solutions is unknown even in 2D. Here we replace the usual CH equation with its physically more relevant nonlocal version. This choice allows us to prove more about the corresponding nonlocal CHHS system. More precisely, we first study wellposedness for the CHB system, endowed with no-slip and no-flux boundary conditions. Then, existence of a weak solution to the CHHS system is obtained as a limit of solutions to the CHB system. Stronger assumptions on the initial datum allow us to prove uniqueness for the CHHS system. Further regularity properties are obtained by assuming additional, though reasonable, assumptions on the interaction kernel. By exploiting these properties, we provide an estimate for the difference between the solution to the CHB system and the one to the CHHS system with respect to viscosity.

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“…For a more general family of two-phase fluid models we also refer the reader to [18,19]. From a modelling perspective the former term is more appropriate in the sense that it can be physically justified and rigorously derived by starting from microscopic models for lattice gases with long-range Kac potentials, see [21,22] and [4,17,20,24,25]. However, the analysis tends to be more challenging and involved since the regularity of ϕ is much lower than in the classical case.…”

Section: Introductionmentioning

confidence: 99%

On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

Gal

2016

J. Math. Fluid Mech.

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Abstract. We consider a diffuse interface model which describes the motion of an ideal incompressible mixture of two immiscible fluids with nonlocal interaction in two-dimensional bounded domains. This model consists of the Euler equation coupled with a convective nonlocal Cahn-Hilliard equation. We establish the existence of globally defined weak solutions as well as well-posedness results for strong/classical solutions.

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Regularity and separation from potential barriers for a non-local phase-field system

Abstract: We show that solutions of a two-phase model involving a non-local interactive term become more regular immediately after the moment they separate from the pure phases. This result allows us to prove stronger convergence to equilibria. A new proof of the separation property is also given.

Cited by 29 publications

References 18 publications

Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems

On an Inviscid Model for Incompressible Two-Phase Flows with Nonlocal Interaction

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