1,721,078 research outputs found
Error estimates for nonlinear Stefan problems obtained as asymptotic limits of a Penrose-Fife model
No full textIn this note I review some results achieved from a collaboration with Jürgen Sprekels and concerning the asymptotic behaviour of initial-boundary value problems for the Penrose-Fife phase-field model as the coefficients of the differential terms for the order parameter tend to zero. In the limit procedure one gets either standard or relaxed Stefan problems, still with heat flux proportional to the gradient of the inverse absolute temperature. After focusing on the problems relaxed in time and recalling the convergence results, we add a small new contribution by proving an error estimate
Global existence for a phase separation system deduced from the entropy balance
No full textThis paper is concerned with a thermomechanical model describing phase separation phenomena in terms of the entropy balance and equilibrium equations for the microforces. The related system is highly nonlinear and admits singular potentials in the phase equation. Both the viscous and the non-viscous cases are considered in the Cahn–Hilliard relations characterizing the phase dynamics. The entropy balance is written in terms of the absolute temperature and of its logarithm, appearing under time derivative. The initial and boundary value problem is considered for the system of partial differential equations. The existence of a global solution is proved via some approximations involving Yosida regularizations and a suitable time discretization
Boundary control problem and optimality conditions for the Cahn–Hilliard equation with dynamic boundary conditions
Full text linkThis paper is concerned with a boundary control problem for the Cahn–Hilliard equation coupled with dynamic boundary conditions. In order to handle the control problem, we restrict our analysis to the case of regular potentials defined on the whole real line, assuming the boundary potential to be dominant. The existence of optimal control, the Fréchet differentiability of the control-to-state operator between appropriate Banach spaces, and the first-order necessary conditions for optimality are addressed. In particular, the necessary condition for optimality is characterised by a variational inequality involving the adjoint variables
Sliding Mode Control for a Generalization of the Caginalp Phase-Field System
No full textIn the present paper, we present and solve the sliding mode control (SMC) problem for a second-order generalization of the Caginalp phase-field system. This generalization, inspired by the theories developed by Green and Naghdi on one side, and Podio-Guidugli on the other, deals with the concept of thermal displacement, i.e., a primitive with respect to the time of the temperature. Two control laws are considered: the former forces the solution to reach a sliding manifold described by a linear constraint between the temperature and the phase variable; the latter forces the phase variable to reach a prescribed distribution φ∗. We prove existence, uniqueness as well as continuous dependence of the solutions for both problems; two regularity results are also given. We also prove that, under suitable conditions, the solutions reach the sliding manifold within finite time
Vanishing diffusion in a dynamic boundary condition for the Cahn–Hilliard equation
No full textThe initial boundary value problem for a Cahn–Hilliard system subject to a dynamic boundary condition of Allen–Cahn type is treated. The vanishing of the surface diffusion on the dynamic boundary condition is the point of emphasis. By the asymptotic analysis as the diffusion coefficient tends to 0, one can expect that the solutions of the surface diffusion problem converge to the solution of the problem without the surface diffusion. This is actually the case, but the solution of the limiting problem naturally looses some regularity. Indeed, the system we investigate is rather complicate due to the presence of nonlinear terms including general maximal monotone graphs both in the bulk and on the boundary. The two graphs are related each to the other by a growth condition, with the boundary graph that dominates the other one. In general, at the asymptotic limit a weaker form of the boundary condition is obtained, but in the case when the two graphs exhibit the same growth the boundary condition still holds almost everywhere
From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation
Full text linkA rigorous proof is given for the convergence of the solutions of a viscous Cahn-Hilliard system to the solution of the regularized version of the forward-backward parabolic equation, as the coefficient of the diffusive term goes to 0. Nonhomogenous Neumann boundary conditions are handled for the chemical potential and the subdifferential of a possible nonsmooth double-well functional is considered in the equation. An error estimate for the difference of solutions is also proved in a suitable norm and with a specified rate of convergence
Remarks on the existence for the one-dimensional Frémond model of shape memory alloys
No full textIn this paper we outline a rigorous proof of the existence of solutions to one-dimensional initial-boundary value problems for the general and complete version of the Frémond thermo-mechanical model applying to shape memory alloys
Convergence to the Stefan problem of the phase relaxation problem with Cattaneo heat flux law
No full textIn this paper we show that the solutions of the phase change problem with the Cattaneo-Fourier heat flux law and phase relaxation, converge to the solution of the Stefan problem as the two relaxation parameters go independently to zero
A Cahn–Hilliard system with forward-backward dynamic boundary condition and non-smooth potentials
Full text linkA system with equation and dynamic boundary condition of Cahn-Hilliard type is considered. This system comes from a derivation performed in Liu-Wu (Arch. Ration. Mech. Anal., 233:167-247, 2019) via an energetic variational approach. Actually, the related problem can be seen as a transmission problem for the phase variable in the bulk and the corresponding variable on the boundary. The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate. Moreover, in the case when the two graphs, in the bulk and on the boundary, exhibit the same growth, we show that the solution of the limit problem is more regular and we prove an error estimate for a suitable order of the diffusion parameter
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