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<p>The Allen–Cahn equation models antiphase domain coarsening in binary mixtures. To capture more complex and natural dynamics, we derive the anisotropic Allen–Cahn (aAC) equation, incorporating anisotropic mobility from an energy functional with a matrix-valued mobility. We present properties of the aAC equation, including energy dissipation and linear stability analysis, using this energy functional. The semi-discretized aAC equation is studied for properties like unique solvability, the maximum principle, and unconditional energy-gradient stability in a time-discrete manner. Additionally, numerical experiments demonstrate properties such as the maximum principle, unconditional energy stability, linear stability analysis, total area decreasing property, and transition layer profile.</p>

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Allen–Cahn equation with matrix-valued anisotropic mobility in two-dimensional space

  • 期刊论文

  • 发表日期：

    2025年1月13日

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  • Gyeonggyu Lee,
  • Seunggyu Lee

摘要

The Allen–Cahn equation models antiphase domain coarsening in binary mixtures. To capture more complex and natural dynamics, we derive the anisotropic Allen–Cahn (aAC) equation, incorporating anisotropic mobility from an energy functional with a matrix-valued mobility. We present properties of the aAC equation, including energy dissipation and linear stability analysis, using this energy functional. The semi-discretized aAC equation is studied for properties like unique solvability, the maximum principle, and unconditional energy-gradient stability in a time-discrete manner. Additionally, numerical experiments demonstrate properties such as the maximum principle, unconditional energy stability, linear stability analysis, total area decreasing property, and transition layer profile.