Calculus of Variations and Geometric Measure Theory

E. Rocca - R. Rossi

``Entropic'' solutions to a thermodynamically consistent PDE system for phase transitions and damage

created by rossi on 18 Mar 2014
modified by rocca on 16 Oct 2015


Published Paper

Inserted: 18 mar 2014
Last Updated: 16 oct 2015

Journal: SIAM J. MATH. ANAL.
Volume: 47
Number: 4
Pages: 2519-2586
Year: 2015


In this paper we analyze a PDE system modelling (non-isothermal) phase transitionsand damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in $L^1$. The whole system has a highly nonlinear character.

We address the existence of a weak notion of solution, referred to as ``entropic'', where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data.

We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its ``entropic'' formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.