DCDS-S
A well-posedness result for irreversible phase transitions with a nonlinear heat flux law
Giovanna Bonfanti Fabio Luterotti
Discrete & Continuous Dynamical Systems - S 2013, 6(2): 331-351 doi: 10.3934/dcdss.2013.6.331
In this paper, we deal with a PDE system describing a phase transition problem characterized by irreversible evolution and ruled by a nonlinear heat flux law. Its derivation comes from the modelling approach proposed by M. Frémond. Our main result consists in showing the global-in-time existence and the uniqueness of the solution of the related initial and boundary value problem.
keywords: existence Phase changes uniqueness. irreversibility microscopic movements
CPAA
Global solution to a phase transition model with microscopic movements and accelerations in one space dimension
Giovanna Bonfanti Fabio Luterotti
Communications on Pure & Applied Analysis 2006, 5(4): 763-777 doi: 10.3934/cpaa.2006.5.763
This note deals with a nonlinear system of PDEs accounting for phase transition phenomena. The existence of solutions of a related Cauchy-Neumann problem is established in the one-dimensional setting. A fixed point procedure guarantees the existence of solutions locally in time. Next, an argument based on a priori estimates allows to extend such solutions in the whole time interval. Hence, the uniqueness of the solution is proved by proper contracting estimates.
keywords: Phase transitions existence and uniqueness results. microscopic movements
DCDS-S
Long-time behaviour of a thermomechanical model for adhesive contact
Elena Bonetti Giovanna Bonfanti Riccarda Rossi
Discrete & Continuous Dynamical Systems - S 2011, 4(2): 273-309 doi: 10.3934/dcdss.2011.4.273
This paper deals with the large-time analysis of a PDE system modelling contact with adhesion, in the case when thermal effects are taken into account. The phenomenon of adhesive contact is described in terms of phase transitions for a surface damage model proposed by M. Frémond. Thermal effects are governed by entropy balance laws. The resulting system is highly nonlinear, mainly due to the presence of internal constraints on the physical variables and the coupling of equations written in a domain and on a contact surface. We prove existence of solutions on the whole time interval $(0,+\infty)$ by a double approximation procedure. Hence, we are able to show that solution trajectories admit cluster points which fulfil the stationary problem associated with the evolutionary system, and that in the large-time limit dissipation vanishes.
keywords: Nonlinear PDE system long-time behaviour. contact with adhesion
DCDS
Analysis of a model coupling volume and surface processes in thermoviscoelasticity
Elena Bonetti Giovanna Bonfanti Riccarda Rossi
Discrete & Continuous Dynamical Systems - A 2015, 35(6): 2349-2403 doi: 10.3934/dcds.2015.35.2349
We focus on a highly nonlinear evolutionary PDE system describing volume processes coupled with surfaces processes in thermoviscoelasticity, featuring the quasi-static momentum balance, the equation for the unidirectional evolution of an internal variable on the surface, and the equations for the temperature in the bulk domain and the temperature on the surface. A significant example of our system occurs in the modeling for the unidirectional evolution of adhesion between a body and a rigid support, subject to thermal fluctuations and in contact with friction.
    We investigate the related initial-boundary value problem, and in particular the issue of existence of global-in-tim solutions, on an abstract level. This allows us to highlight the analytical features of the problem and, at the same time, to exploit the tight coupling between the various equations in order to deduce suitable estimates on (an approximation of) the problem.
    Our existence result is proved by passing to the limit in a carefully tailored approximate problem, and by extending the obtained local-in-time solution by means of a refined prolongation argument.
keywords: maximal monotone operator techniques. global-in-time existence of solutions thermoviscoelasticity volume and surface processes Adhesive contact

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