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Well-posedness of a Pseudo-Parabolic KWC System in Materials Science

Harbir Antil, Daiki Mizuno, Ken Shirakawa

Abstract

The original KWC-system is widely used in materials science. It was proposed in [Kobayashi et al, Physica D, 140, 141--150 (2000)] and is based on the phase field model of planar grain boundary motion. This model suffers from two key challenges. Firstly, it is difficult to establish its relation to physics, in particular, a variational model. Secondly, it lacks uniqueness. The former has been recently studied within the realm of BV-theory. The latter only holds under various simplifications. This article introduces a pseudo-parabolic version of the KWC-system. A direct relationship with variational model (as gradient-flow) and uniqueness are established without making any unrealistic simplifications. Namely, this is the first KWC-system which is both physically and mathematically valid. The proposed model overcomes the well-known open issues.

Well-posedness of a Pseudo-Parabolic KWC System in Materials Science

Abstract

The original KWC-system is widely used in materials science. It was proposed in [Kobayashi et al, Physica D, 140, 141--150 (2000)] and is based on the phase field model of planar grain boundary motion. This model suffers from two key challenges. Firstly, it is difficult to establish its relation to physics, in particular, a variational model. Secondly, it lacks uniqueness. The former has been recently studied within the realm of BV-theory. The latter only holds under various simplifications. This article introduces a pseudo-parabolic version of the KWC-system. A direct relationship with variational model (as gradient-flow) and uniqueness are established without making any unrealistic simplifications. Namely, this is the first KWC-system which is both physically and mathematically valid. The proposed model overcomes the well-known open issues.
Paper Structure (7 sections, 7 theorems, 117 equations)

This paper contains 7 sections, 7 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Approximating method
  5. Proofs of Main Theorems
  6. Proof of Main Theorem 1
  7. Proof of \ref{['mainThm2']}

Key Result

Theorem 1

There exists a sufficiently small constant $\tau_0 \in (0,1)$ such that for any $\tau \in (0,\tau_0)$ and $\varepsilon \in (0,1)$, (AP)$_\tau^\varepsilon$ admits a unique solution $\{ [\eta_i^\varepsilon, \theta_i^\varepsilon] \}_{i = 0}^\infty$. Additionally, the following energy inequality holds:

Theorems & Definitions (15)

  • Definition 1
  • Theorem 1: Solvability of the approximating problem
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • proof : Proof of \ref{['AP']}
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • ...and 5 more