Unified gradient flow structure of phase field systems via a generalized principle of virtual powers

¹ Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy.
elena.bonetti@unimi.it
² Dipartimento di Matematica, Università degli Studi di Pavia, Via Ferrata 5, 27100 Pavia, Italy.
elisabetta.rocca@unipv.it

Received: 14 July 2015
Revised: 5 May 2016
Accepted: 29 June 2016

Abstract

In this paper we introduce a general abstract formulation of a variational thermomechanical model by means of a unified derivation via a generalization of the principle of virtual powers for all the variables of the system, possibly including the thermal one. In particular, through a suitable choice of the driving functional, we formally get a gradient flow structure (in a suitable abstract setting) for the whole nonlinear PDE system. In this framework, the equations may be interpreted as internal balance equations of forces (e.g., thermal or mechanical ones). We prove a global in time existence of (a suitably defined weak) solutions for the Cauchy problem associated to the abstract PDE system as well as uniqueness in case of suitable smoothness assumptions on the functionals.