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

Bonetti, Elena; Rocca, Elisabetta

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 1201-1216

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.

Classification : 74N25, 82B26, 35A01, 35A02
Keywords: Gradient flow, phase field systems, existence of weak solutions, uniqueness

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