Integro-differential equations and functional analysis

Weak solution to KWC systems of pseudo-parabolic type

D. Mizuno

Chiba University, Chiba, Japan

Abstract: In this paper, a class of systems of pseudo-parabolic PDEs is considered. These systems (S)$_\varepsilon$ are derived as a pseudo-parabolic dissipation system of Kobayashi–Warren–Carter energy, proposed by [Kobayashi et al., Physica D, 140, 141–150 (2000)], to describe planar grain boundary motion. In this context, $\varepsilon$ is a value to control the relaxation of singular diffusivity. These systems have been studied in [Antil et al., SIAM J. Math. Anal., 56(5), 6422–6455], and solvability, uniqueness and strong regularity of the solution have been reported under the setting that the initial data is sufficiently smooth. Meanwhile, in this paper, we impose weaker regularity on the initial data, and work on the weak formulation of the systems. In this light, we set our goal of this paper to prove two theorems, concerned with the existence and the uniqueness of weak solution to (S)$_\varepsilon$, and the continuous dependence with respect to the index $\varepsilon$, initial data and forcings.

Keywords: planar grain boundary motion, pseudo-parabolic KWC system, energy-dissipation, singular diffusion, time-discretization.

UDC: 517.9

MSC: 35G61, 35J57, 35J62, 35K70, 74N20

Received: 25.07.2024
Revised: 03.09.2024
Accepted: 06.09.2024

Language: English

DOI: 10.26516/1997-7670.2025.52.88