Pinning of interfaces by localized dry friction

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Courte, Luca¹  Dondl, Patrick¹  Stefanelli, Ulisse^2,3 

[1] Albert Ludwigs Univ Freiburg, Abt Angew Math, Hermann Herder Str 10, D-79104 Freiburg, Germany

[2] Univ Vienna, Fac Math, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria

[3] Ist Matemat Applicata & Tecnol Informat E Magenes, V Ferrata 1, I-27100 Pavia, Italy

关键词: Rate-independent dissipation;    Viscosity solutions;    Random media;    Hysteresis;    Pinning of interfaces;    Equivalence of weak solutions and viscosity solutions;   

DOI  :  10.1016/j.jde.2020.06.005

来源: Elsevier

【 摘 要 】

We consider a model for the evolution of an interface in a heterogeneous environment governed by a parabolic equation. The heterogeneity is introduced as obstacles exerting a localized dry friction. Our main result establishes the emergence of a rate -independent hysteresis for suitable randomly distributed obstacles, i.e., interfaces are pinned by the obstacles until a certain critical applied driving force is exceeded. The treatment of such a model in the context of pinning and depinning requires a comparison principle. We prove this property and hence the existence of viscosity solutions. Moreover, under reasonable assumptions, we show that viscosity solutions are equivalent to weak solutions. ? 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free