Well-posedness and longtime behavior for a singular phase field system with perturbed phase dynamics

Michele Colturato

doi:10.3934/eect.2018011

Abstract

We consider a singular phase field system located in a smooth bounded domain. In the entropy balance equation appears a logarithmic nonlinearity. The second equation of the system, deduced from a balance law for the microscopic forces that are responsible for the phase transition process, is perturbed by an additional term involving a possibly nonlocal maximal monotone operator and arising from a class of sliding mode control problems. We prove existence and uniqueness of the solution for this resulting highly nonlinear system. Moreover, under further assumptions, the longtime behavior of the solution is investigated.

Highlights

This paper is devoted to the mathematical analysis of a system of partial differential equations (PDE) arising from a thermodynamic model describing phase transitions
Suitable physical constraints on χ are introduced: if it is assumed, e.g., that the two phases may coexist at each point with different proportions, it turns out to be reasonable to require that χ lies between 0 and 1, with 1 − χ representing the proportion of the
The second equation of the system under study accounts for the phase dynamics and is deduced from a balance law for the microscopic forces that are responsible for the phase transition process

Summary

Introduction

This paper is devoted to the mathematical analysis of a system of partial differential equations (PDE) arising from a thermodynamic model describing phase transitions. In order to make the presentation clear from the beginning, let us briefly introduce the main ingredients of the PDE system and give some comments on the physical meaning. The unknowns of the problem are the absolute temperature θ and a phase parameter χ which may represent the local proportion of one of the two phases. Suitable physical constraints on χ are introduced: if it is assumed, e.g., that the two phases may coexist at each point with different proportions, it turns out to be reasonable to require that χ lies between 0 and 1, with 1 − χ representing the proportion of the

Objectives

Conclusion