Curvature Effects in Pattern Formation: Well-Posedness and Optimal Control of a Sixth-Order Cahn–Hilliard Equation

IF 2.2 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Mathematical Analysis Pub Date : 2024-07-09 DOI:10.1137/24m1630372

Pierluigi Colli, Gianni Gilardi, Andrea Signori, Jürgen Sprekels

{"title":"Curvature Effects in Pattern Formation: Well-Posedness and Optimal Control of a Sixth-Order Cahn–Hilliard Equation","authors":"Pierluigi Colli, Gianni Gilardi, Andrea Signori, Jürgen Sprekels","doi":"10.1137/24m1630372","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4928-4969, August 2024. <br/> Abstract. This work investigates the well-posedness and optimal control of a sixth-order Cahn–Hilliard equation, a higher-order variant of the celebrated and well-established Cahn–Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improving the understanding of the mathematical properties and control aspects of the sixth-order Cahn–Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Mathematical Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/24m1630372","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4928-4969, August 2024.
Abstract. This work investigates the well-posedness and optimal control of a sixth-order Cahn–Hilliard equation, a higher-order variant of the celebrated and well-established Cahn–Hilliard equation. The equation is endowed with a source term, where the control variable enters as a distributed mass regulator. The inclusion of additional spatial derivatives in the sixth-order formulation enables the model to capture curvature effects, leading to a more accurate depiction of isothermal phase separation dynamics in complex materials systems. We provide a well-posedness result for the aforementioned system when the corresponding nonlinearity of double-well shape is regular and then analyze a corresponding optimal control problem. For the latter, existence of optimal controls is established, and the first-order necessary optimality conditions are characterized via a suitable variational inequality. These results aim at contributing to improving the understanding of the mathematical properties and control aspects of the sixth-order Cahn–Hilliard equation, offering potential applications in the design and optimization of materials with tailored microstructures and properties.

查看原文本刊更多论文

图案形成中的曲率效应：六阶卡恩-希利亚德方程的拟合与优化控制

SIAM 数学分析期刊》，第 56 卷第 4 期，第 4928-4969 页，2024 年 8 月。摘要本研究探讨了六阶 Cahn-Hilliard 方程的良好求解性和最优控制，该方程是著名且成熟的 Cahn-Hilliard 方程的高阶变体。该方程带有一个源项，控制变量以分布式质量调节器的形式进入。在六阶公式中加入额外的空间导数，使模型能够捕捉曲率效应，从而更准确地描述复杂材料系统中的等温相分离动力学。当相应的双井形状非线性有规律时，我们给出了上述系统的好求结果，然后分析了相应的最优控制问题。对于后者，我们建立了最优控制的存在性，并通过适当的变分不等式描述了一阶必要最优条件。这些结果旨在加深对六阶 Cahn-Hilliard 方程的数学特性和控制方面的理解，为设计和优化具有定制微结构和特性的材料提供潜在应用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

SIAM Journal on Mathematical Analysis 数学-应用数学

CiteScore

3.30

自引率

5.00%

发文量

175

审稿时长

12 months

期刊介绍： SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.