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<p>In this paper, we study the initial-boundary value problem for a one-dimensional parabolic equation with a non-constant Hölder diffusion coefficient and a rough forcing term. This type of special problem originates from the solving process of the Ericksen–Leslie model for the Poiseuille flow of nematic liquid crystals. By adopting the parametrix method to construct the Green function and using the mollification technique to mollify the rough term, we establish the global existence and Hölder regularity of weak solutions to the initial-boundary value problem. It is also verified that the Hölder exponent of the solution is explicitly related to that of the diffusion coefficient.</p>

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The initial-boundary value problem to a parabolic equation with Hölder continuous diffusion coefficient

  • 期刊论文

  • 发表日期：

    2026年6月2日

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  • Linyang Yuan,
  • Yanbo Hu

摘要

In this paper, we study the initial-boundary value problem for a one-dimensional parabolic equation with a non-constant Hölder diffusion coefficient and a rough forcing term. This type of special problem originates from the solving process of the Ericksen–Leslie model for the Poiseuille flow of nematic liquid crystals. By adopting the parametrix method to construct the Green function and using the mollification technique to mollify the rough term, we establish the global existence and Hölder regularity of weak solutions to the initial-boundary value problem. It is also verified that the Hölder exponent of the solution is explicitly related to that of the diffusion coefficient.