Kobayashi–Warren–Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation. (April 2022)

• Record Type: Journal Article
• Authors: Moll, Salvador; Shirakawa, Ken; Watanabe, Hiroshi
• Is Part Of: Nonlinear analysis. Volume 217(2022)
• Publication Date: 2022-04
• Subjects: 35K87 -- 35R06 -- 35K67; Parabolic system -- Grain boundary motion -- ω-limit -- Energy dissipation -- SBV-functions; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2021.112722
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we study the Dirichlet problem for the Kobayashi–Warren–Carter system. This system of parabolic PDE's models the grain boundary motion in a polycrystal with a prescribed orientation at the boundary of the domain. We obtain global existence in time of energy-dissipative solutions. The regularity of the solutions as well as the energy-dissipation property permit us to derive the steady-state problem as the asymptotic in time limit of the system. We finally study the ω -limit set of the solutions; we completely characterize it in the one dimensional case, showing, in particular that orientations in the ω -limit set belong to the space of SBV functions. In the two dimensional case, we give sufficient conditions for existence of radial symmetric piecewise constant solutions.