Diffuse interface model for two-phase flows on evolving surfaces with different densities: global well-posedness.

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2025-01-01 Epub Date: 2025-05-04 DOI:10.1007/s00526-025-03001-w

Helmut Abels, Harald Garcke, Andrea Poiatti

引用次数: 0

Abstract

We show global in time existence and uniqueness on any finite time interval of strong solutions to a Navier-Stokes/Cahn-Hilliard type system on a given two-dimensional evolving surface in the case of different densities and a singular (logarithmic) potential. The system describes a diffuse interface model for a two-phase flow of viscous incompressible fluids on an evolving surface. We also establish the validity of the instantaneous strict separation property from the pure phases. To show these results we use our previous achievements on local well-posedness together with suitable novel regularity results for the convective Cahn-Hilliard equation. The latter allows to obtain higher-order energy estimates to extend the local solution globally in time. To this aim the time evolution of energy type quantities has to be calculated and estimated carefully.

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不同密度演化表面上两相流的扩散界面模型：全局适定性。

我们证明了给定二维演化曲面上具有不同密度和奇异（对数）势的Navier-Stokes/Cahn-Hilliard型系统的强解在任意有限时间区间上的全局存在唯一性。该系统描述了粘性不可压缩流体在不断变化的表面上的两相流动的扩散界面模型。我们还建立了纯相的瞬时严格分离性质的有效性。为了证明这些结果，我们使用了我们以前关于局部适定性的成果，以及关于对流Cahn-Hilliard方程的合适的新正则性结果。后者允许获得高阶能量估计，从而及时将局部解扩展到全局。为此，必须仔细计算和估计能量型量的时间演化。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.