Phase-field systems with nonlinear coupling and dynamic boundary conditions

Cavaterra, Cecilia; Gal, Ciprian G.; Grasselli, Maurizio; Miranville, Alain

doi:10.1016/j.na.2009.11.002

Cited by 30 publications

(36 citation statements)

References 38 publications

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“…The second reason involves the presence of the dynamic flux term ∂ n u t in our boundary condition. Recently, works such as [8,18,19,20,21,27,33] are able to establish the existence of an exponential attractor through for wave equations with dynamic boundary conditions through the use of suitable H 2 -elliptic regularity estimates. Because of the dynamic flux term, ∂ n u t , such estimates are not available here.…”

Section: Thenζ Andū Satisfy the Equationsmentioning

confidence: 99%

Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions

Graber

Shomberg

2016

Nonlinearity

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Abstract. We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the presence of a parameter distinguishing whether the underlying operator is analytic, α > 0, or only of Gevrey class, α = 0. We establish the existence of a global attractor for each α ∈ [0, 1], and we show that the family of global attractors is upper-semicontinuous as α → 0. Furthermore, for each α ∈ [0, 1], we show the existence of a weak exponential attractor. A weak exponential attractor is a finite dimensional compact set in the weak topology of the phase space. This result insures the corresponding global attractor also possess finite fractal dimension in the weak topology; moreover, the dimension is independent of the perturbation parameter α. In both settings, attractors are found under minimal assumptions on the nonlinear terms.

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Section: Thenζ Andū Satisfy the Equationsmentioning

confidence: 99%

Attractors for strongly damped wave equations with nonlinear hyperbolic dynamic boundary conditions

Graber

Shomberg

2016

Nonlinearity

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“…Indeed, the map Ξ : u → A Γ u, when viewed as a map from V 2 into L 2 (Ω) × L 2 (Γ ), is an isomorphism and, cf., e.g., [3], Lemma 2.2, there exists a positive constant C * , independent of (u, ψ), such that…”

Section: Variational Formulation and Well-posednessmentioning

confidence: 99%

“…Finally, we consider the variational identity (3.8). On account of the regularity properties of ∂ t u, ∂ t ψ and μ, we can use the results in [40], Appendix A (see also [3], Lemma 2.2) to infer that (u, ψ) ∈ L ∞ (J; V 3 ). This concludes the proof.…”

Section: Cavaterra Et Al / Cahn-hilliard Equations With Memory Anmentioning

confidence: 99%

Cahn–Hilliard equations with memory and dynamic boundary conditions

Cavaterra

Gal

Grasselli

2011

Asymptotic Analysis

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We consider a modified Cahn-Hiliard equation where the velocity of the order parameter u depends on the past history of Δμ, μ being the chemical potential with an additional viscous term αu t , α 0. This type of equation has been proposed by P. Galenko et al. to model phase separation phenomena in special materials (e.g., glasses). In addition, the usual noflux boundary condition for u is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The resulting boundary value problem is subject to suitable initial conditions and is reformulated in the socalled past history space. Existence of a variational solution is obtained. Then, in the case α > 0, we can also prove uniqueness and construct a strongly continuous semigroup acting on a suitable phase space. We show that the corresponding dynamical system has a (smooth) global attractor as well as an exponential attractor. In the case α = 0, we only establish the existence of a trajectory attractor.(1.6) Equation (1.6) has received lot of attention in the last years. The viscous three-dimensional case has been firstly analyzed in [21] (see also [34,35]). It is worth observing that, in this case, the solutions regularize C. Cavaterra et al. / Cahn-Hilliard equations with memory and dynamic boundary conditions 125in finite time (as in the classical case = 0). The nonviscous case is much more troublesome even in dimension two, since there is a lack of regularization effects (a sort of "hyperbolic" behavior). Only recently, the understanding of the case α = 0 has significantly improved (see [28][29][30]). For the simpler one-dimensional case, the reader is referred to [1] and its references. Equation (1.5) with a more general memory kernel (which does not allow to write an equivalent PDE) has been firstly studied in [22] in the one dimensional nonviscous case (see also [23] for no-flux boundary conditions). The corresponding dynamical system has been analyzed within the past history setting with particular regard to the stability with respect to the relaxation time. The three-dimensional nonisothermal nonviscous case has been considered in [39] and the existence of a (weak) solution has been proven. In the related paper [53], well-posedness and regularity results have been obtained by taking memory kernels which are singular at 0. We recall that this assumption, contrary to nonsingular kernels, produces regularization effects. More recently, Eq. (1.6) with α > 0 has been carefully analyzed in the spirit of [21]. This means that the kernel k has been rescaled with a relaxation time > 0 and the robustness properties of the dynamical system with respect to α and have been investigated. In particular, the authors have constructed a family of exponential attractors which is continuous as (α, ) goes to (0, 0), provided that is dominated by α. Of course, the latter restriction is expected due to the difficulties arising in the relaxed nonviscous case. In other words, the dynamical system (in the history phase space) genera...

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“…They have been used, among others, as a model of "boundary feedback" in stabilization and control problems of membranes and plates, [3,13,14,12,15,23], in phase transition problems, [22,7,8,9,17,4], in some hydrodynamic problems, [10,21] or in population dynamics, [6]. They have also been considered in the context of ellipticparabolic problems, [5,18].…”

Section: Introductionmentioning

confidence: 99%