含表面活性剂扩散界面两相流模型的局部适定性
Local Well-posedness of a Diffuse Interface Two-phase Flow Model with Surfactants
- Universität Regensburg（雷根斯堡大学）
- Anhui Normal University（安徽师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对含表面活性剂两相流的扩散界面模型C，通过定制预解分析与算子值Mikhlin乘子定理克服混合阶强耦合困难，建立强解局部适定性，并推广至结构解耦的模型D。
AI中文摘要：
我们建立了描述含表面活性剂两相流的热力学一致扩散界面模型的强解局部适定性，具体而言是Garcke、Lam和Stinner引入的所谓模型C。由于模型C具有非标准、混合阶强耦合结构，经典Agmon-Douglis-Nirenberg理论和常规Lopatinskii-Shapiro边界条件因发散标度行为而不适用。为克服此困难，我们在合适的Hilbert空间中发展了一种针对源项时间正则性定制的预解分析，该分析在估计中实现了最高阶交叉耦合的精确抵消。这一关键解耦使得算子值Mikhlin乘子定理得以严格应用，从而在$\mathbb{R}$上获得预解估计。此外，我们分析了一个结构变体，记为模型D，该模型通过将界面表面活性剂能量完全从梯度部分转移到势能部分，实现了关键的结构解耦，并随即建立了强解的局部适定性。
英文摘要：
We establish the local well-posedness of strong solutions for a thermodynamically consistent diffuse interface model describing two-phase flows with surfactants, specifically the so-called Model C introduced by Garcke, Lam, and Stinner. Because of the non-standard, mixed-order strongly coupled structure of Model C, the classical Agmon-Douglis-Nirenberg theory and conventional Lopatinskii-Shapiro boundary conditions are not applicable due to divergent scaling behaviors. To overcome this, we develop a tailored resolvent analysis, adapted to the temporal regularity of the source term, in a suitable Hilbert space, which yields an exact cancellation of the highest-order cross-couplings in the estimates. This crucial decoupling enables the rigorous application of the operator-valued Mikhlin multiplier theorem to obtain resolvent estimates on $\mathbb{R}$. Furthermore, we analyze a structural variant, denoted as Model D, which achieves a crucial structural decoupling by transferring the interfacial surfactant energy entirely from the gradient part to the potential part, and we readily establish local well-posedness for strong solutions.
