Spherically symmetric solutions to a model for phase transitions driven by configurational forces

TL;DR

This paper proves the global existence of spherically symmetric solutions for a PDE system modeling phase transitions driven by configurational forces, simplifying the problem to one dimension.

Contribution

It establishes the existence of solutions for a complex PDE system modeling martensitic phase transitions with spherical symmetry, extending the understanding of such models.

Findings

Proved global in time existence of solutions.

Reduced multidimensional problem to one dimension.

Validated the approach for spherically symmetric cases.

Abstract

We prove the global in time existence of spherically symmetric solutions to an initial-boundary value problem for a system of partial differential equations, which consists of the equations of linear elasticity and a nonlinear, non-uniformly parabolic equation of second order. The problem models the behavior in time of materials in which martensitic phase transitions, driven by configurational forces, take place, and can be considered to be a regularization of the corresponding sharp interface model. By assuming that the solutions are spherically symmetric, we reduce the original multidimensional problem to the one in one space dimension, then prove the existence of spherically symmetric solutions. Our proof is valid due to the essential feature that the reduced problem is one space dimensional.