Kobayashi–Warren–Carter type systems with nonhomogeneous Dirichlet boundary data for crystalline orientation. (April 2022) - Moll, Salvador; Shirakawa, Ken; Watanabe, Hiroshi
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.