Existence of solutions for a mathematical model related to solid-solid phase transitions in shape memory alloys

TL;DR

This paper proves the existence of weak solutions for a complex nonlinear PDE system modeling solid-solid phase transitions in shape memory alloys, involving coupled temperature, stress, and phase evolution equations.

Contribution

It introduces a novel approach to handle the nonlinear coupling, multivalued operators, and low regularity data in the PDE system for shape memory alloys.

Findings

Existence of weak solutions for the PDE system.

Development of a regularization and discretization method.

Establishment of a priori estimates for passing to the limit.

Abstract

We consider a strongly nonlinear PDE system describing solid-solid phase transitions in shape memory alloys. The system accounts for the evolution of an order parameter (related to different symmetries of the crystal lattice in the phase configurations), of the stress (and the displacement), and of the absolute temperature. The resulting equations present several technical difficulties to be tackled: in particular, we emphasize the presence of nonlinear coupling terms, higher order dissipative contributions, possibly multivalued operators. As for the evolution of temperature, a highly nonlinear parabolic equation has to be solved for a right hand side that is controlled only in L^1. We prove the existence of a solution for a regularized version, by use of a time discretization technique. Then, we perform suitable a priori estimates which allow us pass to the limit and find a weak global-in-time solution to the system.