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A nonisothermal phase field system of Cahn-Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It…

Analysis of PDEs · Mathematics 2023-12-15 Pierluigi Colli , Gianni Gilardi , Andrea Signori , Jürgen Sprekels

We consider a non-isothermal modified Cahn--Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation.…

Analysis of PDEs · Mathematics 2013-10-04 Cecilia Cavaterra , Maurizio Grasselli , Hao Wu

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The…

Optimization and Control · Mathematics 2023-03-02 Pierluigi Colli , Gianni Gilardi , Andrea Signori , Jürgen Sprekels

In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It…

Optimization and Control · Mathematics 2023-03-24 Pierluigi Colli , Gianni Gilardi , Andrea Signori , Jürgen Sprekels

We consider a differential model describing nonisothermal fast phase separation processes taking place in a three-dimensional bounded domain. This model consists of a viscous Cahn-Hilliard equation characterized by the presence of an…

Analysis of PDEs · Mathematics 2007-05-23 Maurizio Grasselli , Hana Petzeltova , Giulio Schimperna

This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order…

Analysis of PDEs · Mathematics 2011-09-16 Pierluigi Colli , Gianni Gilardi , Paolo Podio-Guidugli , Jürgen Sprekels

P. Galenko et al. proposed a modified Cahn-Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on…

Analysis of PDEs · Mathematics 2008-04-08 Maurizio Grasselli , Giulio Schimperna , Sergey Zelik

An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions.…

Analysis of PDEs · Mathematics 2012-02-24 Pierluigi Colli , Gianni Gilardi , Paolo Podio-Guidugli , Jürgen Sprekels

We study a system describing the dynamics of a two-phase flow of incompressible viscous fluids influenced by the convective heat transfer of Caginalp-type. The separation of the fluids is expressed by the order parameter which is of diffuse…

Analysis of PDEs · Mathematics 2023-08-15 Šárka Nečasová , John Sebastian H. Simon

We are concerned with a phase field system consisting of two partial differential equations in terms of the variables thermal displacement, that is basically the time integration of temperature, and phase parameter. The system is a…

Analysis of PDEs · Mathematics 2012-05-29 Giacomo Canevari , Pierluigi Colli

We study a diffusion model of phase field type, consisting of a system of two partial differential equations encoding the balances of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a…

Analysis of PDEs · Mathematics 2011-03-24 Pierluigi Colli , Gianni Gilardi , Paolo Podio-Guidugli , Juergen Sprekels

This work explores the solvability of a sixth-order Cahn--Hilliard equation with an inertial term, which serves as a relaxation of a higher-order variant of the classical Cahn--Hilliard equation. The equation includes a source term that…

Analysis of PDEs · Mathematics 2025-04-14 Pierluigi Colli , Gianni Gilardi

A common paradigm in phase-field models with singular potentials is that global-in-time weak solutions converge to a single equilibrium only after undergoing asymptotic regularization. However, in arXiv:2510.17296 we introduced a novel…

Analysis of PDEs · Mathematics 2026-04-01 Maurizio Grasselli , Andrea Poiatti

We propose a new class of phase field models coupled to viscoelasticity with large deformations, obtained from a diffuse interface mixture model composed by a phase with elastic properties and a liquid phase. The model is formulated in the…

Analysis of PDEs · Mathematics 2022-04-12 Abramo Agosti , Pierluigi Colli , Harald Garcke , Elisabetta Rocca

In this paper we deal with a singular nonlocal phase field system with inertial term. The system has the logarithm of the absolute temperature $\theta$ under time derivative. Although the system has a difficult mathematical point caused by…

Analysis of PDEs · Mathematics 2021-05-18 Shunsuke Kurima

Isothermal compressible two-phase flows with and without phase transition are modeled, employing Darcy's and/or Forchheimer's law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense…

Analysis of PDEs · Mathematics 2018-07-09 Jan Pruess , Gieri Simonett

Phase-field models are a popular choice in computational physics to describe complex dynamics of substances with multiple phases and are widely used in various applications. We present nonlocal non-isothermal phase-field models of…

Analysis of PDEs · Mathematics 2023-10-13 Olena Burkovska

Phase field models are widely used to describe multiphase systems. Here a smooth indicator function, called phase field, is used to describe the spatial distribution of the phases under investigation. Material properties like density or…

Numerical Analysis · Mathematics 2015-11-10 Christian Kahle

This paper is concerned with a non-isothermal Cahn-Hilliard model based on a microforce balance. The model was derived by A. Miranville and G. Schimperna starting from the two fundamental laws of Thermodynamics, following M. Gurtin's…

Analysis of PDEs · Mathematics 2020-04-07 Alice Marveggio , Giulio Schimperna

A Cahn-Hilliard-Allen-Cahn phase-field model coupled with a heat transfer equation, particularly with full non-diagonal mobility matrices, is studied. After reformulating the problem w.r.t. the inverse of temperature, we proposed and…

Numerical Analysis · Mathematics 2024-08-01 Aaron Brunk , Oliver Habrich , Timileyin David Oyedeji , Yangyiwei Yang , Bai-Xiang Xu
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