[论文解读] Energy-variational solutions for geodynamical two-phase flows -- From logarithmic to double-obstacle potentials by variational convergence

Cheng Fan, Robert Lasarzik+1 more|arXiv (Cornell University)|Jan 13, 2026

Solidification and crystal growth phenomena被引用 0

一句话总结

该论文为地质动力学两相流引入能量变分解并分析对数势到双障势能的变分极限，凸显其相较耗散解的优势。

ABSTRACT

In [Cheng, Lasarzik, Thomas 2025 ARXIV-Preprint 2509.25508], we studied a Cahn--Hilliard two-phase model describing the flow of two viscoelastoplastic fluids in the framework of dissipative solutions using a logarithmic potential for the phase-field variable. This choice of potential has the effect that the fluid mixture cannot fully separate into two pure phases. The notion of dissipative solutions is based on a relative energy-dissipation inequality featuring a suitable regularity weight. In this way, this is a very weak solution concept. In the present work, we study the well-posedness of the geodynamical two-phase flow in the notion of energy-variational solutions. They feature an additional scalar energy variable that majorizes the system energy along solutions and they are further characterized by a variational inequality that combines an energy-dissipation estimate with the weak formulation of the system adding an error term that accounts for the mismatch between the energy variable and the system energy multiplied by a suitable regularity weight. We give a comparison of these two concepts. We further study different phase-field potentials for the geodynamical two-phase flow model. In particular, we address the variational limit from a potential with a logarithmic contribution to a double-obstacle potential, then also allowing for the emergence of pure phases. This study underlines that, thanks to its structure, the energy-variational solution is better suited for variational convergence methods than the dissipative solution.

研究动机与目标

• 用扩展介面的不可压黏弹塑性流体的两相地质动力学流动进行动机化建模。
• 通过包含辅助能量和正则性加权误差项的能量-变分解建立良定性框架。
• 研究从对数相场势到双障势势的变分收敛及其对尖界极限的影响。
• 将能量-变分解与耗散解比较并阐明变分收敛方法的优势。

提出的方法

• 给出一个以辅助能量变量E(t)为主⼤的能量-变分解定义，使其覆盖系统能量。
• 引入正则性权重K，在能量平衡中得到一个下半连续的缺陷项。
• 使用伽马极限（Gamma-convergence）和图极限（graph-convergence）在弱收敛的逼近序列中通过极限。
• 对地质动力学两相系统应用时间离散方案，在存在应力扩散（γ>0）的情况下证明存在性。
• 令γ→0得到非正则化体系的能量-变分解极限。
• 在Cahn–Hilliard 成分中将对数势到双障势进行α→0的变分极限，并分析得到的能量-变分解。

实验结果

研究问题

• RQ1 如何为具有非光滑耗散的地质动力学两相流形式化能量-变分解？
• RQ2 在弱极限下，能量-变分框架是否相较于耗散解提供更强的收敛性？
• RQ3 从对数相场势到双障势势的变分极限对解的性质和收敛性有何影响？
• RQ4 在变分极限中是否能够回收纯相或允许相边界达到纯态？
• RQ5 变分极限与模型中的非恒定迁移率和应力扩散之间有何相互作用？

主要发现

• 能量-变分解始终具备耗散性并满足半流动性质。
• 对于含对数相场势的系统建立了存在性结果。
• 通过γ→0极限获得不含应力扩散的能量-变分解。
• 在能量-变分框架内分析了从对数势到双障势的α→0变分极限。
• 该方法强调能量-变分解非常适用于变分收敛方法，与耗散解相比具有优势。