On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy–Neumann and nonlinear dynamic boundary conditions

Miranville, Alain; Moroşanu, Costică

doi:10.1016/j.apm.2015.04.039

Cited by 16 publications

(18 citation statements)

References 11 publications

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“…Proof. Applying Lemma 2.3 in [26] with h 3 = −w and making use of the embeddings (2)), we can easily conclude that the results set out by Lemma 2.1 are true.…”

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confidence: 73%

“…On the existence, regularity, stability and uniqueness of solutions to the phase field nonlinear equation with non-homogeneous dynamic boundary conditions. In order to study the nonlinear problem (1) 1 , we will appeal to the strategy used in [26]. In this sense, we will consider a further variable ψ = ϕ, ψ(0, x) = ϕ 0 on Γ and we will treat the dynamic boundary conditions (1) 2 as a parabolic equation for ψ on the boundary, i.e.…”

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confidence: 99%

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Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions

Miranville¹,

Moroşanu²

2016

DCDS-S

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The paper concerns with the existence, uniqueness, regularity and the approximation of solutions to the nonlinear phase-field (Allen-Cahn) equation, endowed with non-homogeneous dynamic boundary conditions (depending both on time and space variables). It extends the already studied types of boundary conditions, which makes the problem to be more able to describe many important phenomena of two-phase systems, in particular, the interactions with the walls in confined systems. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual numerical algorithm is formulated in the end.2010 Mathematics Subject Classification. Primary: 35K55; Secondary: 65N12, 80AXX. Key words and phrases. Boundary value problems for nonlinear parabolic PDE, stability and convergence of numerical methods, nonhomogeneous dynamic boundary conditions, thermodynamics, heat transfer. 537 538 ALAIN MIRANVILLE AND COSTICȂ MOROŞANUThe equation (1) 1 was introduced initially by Allen-Cahn (see [1]) to describe the motion of anti-phase boundaries in crystalline solids. Recently, the Allen-Cahn equation has been widely applied to many complex moving interface problems, like: the mixture of two incompressible fluids, the nucleation of solids, vesicle membranes, etc. Also, the nonlinear parabolic equation (1) 1 occurs in the Caginalp's phase-field transition system (see [6]) describing the transition between the solid and liquid phases in the solidification process of a material occupying a region Ω.The main novelty of this paper refer to the presence of non-homogeneous dynamic boundary conditions in equation (1), untreated until now (to our knowledge) in the mathematical literature, which makes the present nonlinear parabolic problem to be more able describing many important phenomena of two-phase systems: superheating, supercooling, the effects of surface tension, separating zones etc, in particular, the interactions with the walls in confined systems. Consequently, a wide variety of industrial applications are covered. Endowed with homogeneous dynamic boundary conditions and singular potentials, problem (1) was treated in [7], [10]-[12] and [19]. For detailed discussions on the phase-field transition system we refer to [5]-[13], [15]-[20], [23], [30]-[33].At the moment t the material is considered to be liquid if ϕ is close to 1 + δ 1 , while it is considered to be solid if ϕ is close to −1 − δ 1 , with δ 1 a positive number. Let us consider the interface as a continuous region, more vast (in which the liquid can coexist with the solid), of finite thickness, in which the change of phase occurs continuously. In this spirit we define the separating region (the interface at the moment t) as being the set:Here we will use the standard notations for Sobolev spaces, namely, given positive integer k and 1 ≤ p ≤ ∞, we denote by W k,2...

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“…Proof. Applying Lemma 2.3 in [26] with h 3 = −w and making use of the embeddings (2)), we can easily conclude that the results set out by Lemma 2.1 are true.…”

mentioning

confidence: 73%

mentioning

confidence: 99%

Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions

Miranville¹,

Moroşanu²

2016

DCDS-S

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“…the nonlinearities in (1.1) become F (ϕ) = 1 2ξ (ϕ − ϕ 3 ) (see also [26]). Further, the regular potential indicated by (1.3) also includes the general nonlinearities…”

Section: Introductionmentioning

confidence: 97%

“…Consequently, a wide variety of industrial applications are covered. For detailed discussions on the phase-field transition system we refer to [12][13][14][15][16][17]20,22,26,27,[30][31][32]34]. In [33] the reader can found more details relative to a more extensive class of problems on the type those treated in this paper (reaction-diffusion equation), as well as different types for the nonlinear term F (ϕ).…”

Section: Introductionmentioning

confidence: 99%

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

Moroşanu

Croitoru

2015

Journal of Mathematical Analysis and Applications

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Keywords:Boundary value problems for nonlinear parabolic PDE Stability and convergence of numerical method Thermodynamics Heat transferThe paper studies the existence, uniqueness, regularity and the approximation of solutions to the reaction-diffusion equation endowed with a general nonlinear regular potential and Cauchy-Neumann boundary conditions. The convergence and error estimate results for an iterative scheme of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation necessary to be done in order to approximate the solution of nonlinear parabolic equation.

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“…The classical regular potential in Caginalp's model (see [6]) is obtained for p 3 = 1 2ξ , i.e. the nonlinearities in (1) becomes F (ϕ) = 1 2ξ (ϕ − ϕ 3 ) (see also [22]). Further, the regular potential indicated above also includes the general nonlinearities F (z) = a 1 z +a 2 z 2 +...+a 2p−2 z 2p−2 +a 2p−1 z 2p−1 with a 2p−1 < 0 (see [28] and references therein).…”

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Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation

Moroşanu¹

2018

Discrete &Amp; Continuous Dynamical Systems - S

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We present the error analysis of two time-stepping schemes of fractional steps type, used in the discretization of a nonlinear reaction-diffusion equation with Neumann boundary conditions, relevant in phase transition and interface problems. We start by investigating the solvability of a such boundary value problems in the class W 1,2 p (Q). One proves the existence, the regularity and the uniqueness of solutions, in the presence of the cubic nonlinearity type. The convergence and error estimate results (using energy methods) for the iterative schemes of fractional steps type, associated to the nonlinear parabolic equation, are also established. The advantage of such method consists in simplifying the numerical computation. On the basis of this approach, a conceptual algorithm is formulated in the end. Numerical experiments are presented in order to validates the theoretical results (conditions of numerical stability) and to compare the accuracy of the methods.

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On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy–Neumann and nonlinear dynamic boundary conditions

Cited by 16 publications

References 11 publications

Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions

Analysis of an iterative scheme of fractional steps type associated to the nonlinear phase-field equation with non-homogeneous dynamic boundary conditions

Analysis of an iterative scheme of fractional steps type associated to the reaction–diffusion equation endowed with a general nonlinearity and Cauchy–Neumann boundary conditions

Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation

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