Regularity of Solutions to a Model for Solid-Solid Phase Transitions Driven by Configurational Forces

Sep 4, 2026

Regularity results are developed for a model of solid evolution with martensitic phase transitions, where phase interfaces are treated as diffusive and the driving mechanism is configurational forces. Building on prior existence of weak solutions for an initial-boundary value problem, the study upgrades the initial data to H2(Ω) and analyzes weak-solution regularity, with difficulty arising from a gradient-weight term in the principal part. The theory is proved only in one spatial dimension.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages20

Reading Time50 min

AccessFree to download

Introduction

Phase-field vs. sharp-interface modeling

Temporal evolution equations (Cahn–Hilliard/Allen–Cahn)

Diffusive-interface model driven by configurational forces

Problem formulation and one-dimensional reduction

Regularity under H2(Ω) initial data

Regularity of solutions to a model for solid-solid phase transitions driven by con􀀌gurational forces Peicheng Zhu 1 ;2 􀀃 1 Basque Center for Applied Mathematics (BCAM) Building 500, Bizkaia Technology Park E-48160 Derio, Spain 2 IKERBASQUE, Basque Foundation for Science E-48011 Bilbao, Spain Abstract In a previous work, we prove the existence of weak solutions to an initial-boundary value problem, with H 1 (Ω) initial data, for a system of partial diﬀerential equations, which consists of the equations of linear elasticity and a nonlinear, degenerate parabolic equation of second order. Assuming in this article the initial data is in H 2 (Ω), we investigate the regularity of weak solutions that is diﬃcult due to the gradient term which plays a role of a weight. The problem models the behavior in time of materials with martensitic phase transitions. This model with diﬀusive phase interfaces was derived from a model with sharp interfaces, whose evolution is driven by conﬁgurational forces, and can be thought to be a regularization of that model. Our proof, in which the diﬃculties are caused by the weight in the principle term, is only valid in one space dimension. Keywords. Regularity, Weak solutions, Elliptic-parabolic system, Phase transition model MSC 2000 . 35K55; 74E15 . 1 Introduction Many inhomogeneous systems can be characterized by domains of diﬀerent phases separated by a distinct interface [16] . Driven out of equilibrium, their dynamics result in the evolution of those interfaces which might develop into structures (compositional and structural inhomogeneities) with characteristic length scales at the nano-, micro-or mesoscale. To a large extent, the material properties of such systems are determined by those structures of small-scale. It is thus important to understand precisely the mechanisms that drive the evolution of those structures. In this article we are interested in a model for the evolution, driven by conﬁgurational forces, of microstructures in elastically deformable solids. Materials microstructures may consist of spatially distributed phases of diﬀerent compositions and/or crystal structures, grains of diﬀerent orientations, domains of diﬀerent structural variants, domains of diﬀerent electrical or magnetic polarizations, and structural defects, see e.g. [15] . These structural features usually have an intermediate mesoscopic length scale in the range of nanometers to microns. The size, shape, and ∗ E-mail: [zhu@bcamath.org](zhu@bcamath.org) spatial arrangement of the local structural features in a microstructure play a critical role in determining the physical properties of a material. Because of the complex and nonlinear nature of microstructure evolution, numerical approaches are often employed. There are two main types of modeling for the evolution of microstructures. In the conventional approach, the regions separating the domains are treated as mathematically sharp interfaces. The local interfacial velocity is then determined as part of the boundary conditions, or is calculated from the driving force for interface motion and the interfacial mobility. This approach requires the explicit tracking of the interface positions. Such an interface-tracking approach can be successful in one-dimensional systems, however it will be impractical for complicated three-dimensional microstructures. Therefore, during the past decades, another approach has been invented, namely, the phase-ﬁeld approach in which the interface is not of zero thickness, instead an interfacial region with thickness of certain order of a small regularization parameter. Though it is still a young discipline in condensed matter physics, this approach has emerged to be one of the most powerful methods for modeling the evolution of microstructures. It can be traced back the theory of diﬀuse-interface description, which is developed, independently, more than a century ago by van der Waals [27] and some half centur

FAQs

What mathematical objects are studied in the paper?

Weak solutions of a coupled elliptic–parabolic system describing linear elasticity coupled to a quasilinear, non-uniformly parabolic second-order equation for diffusive phase transitions.

Why is regularity difficult in this model?

The gradient term acts as a weight in the principal part, creating technical difficulties for proving regularity of weak solutions.

What assumptions are made on the initial data to obtain the main regularity results?

The paper assumes initial data belongs to H2(Ω), extending earlier work that required only H1(Ω) initial data for existence.

Sep 4, 2026

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