Error estimates of effective boundary conditions for the heat equation with optimally aligned coatings

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Mathematical Analysis and Applications Pub Date : 2024-10-16 DOI:10.1016/j.jmaa.2024.128972

Lixin Meng , Zhitong Zhou

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Abstract

We are interested in the validity of effective boundary conditions for a heat equation on a coated body as the thickness of the coating shrinks to zero. The coating is optimally aligned in the sense that the normal vector in the coating is an eigenvector of the thermal tensor. If the heat equation satisfies Neumann boundary condition on the outer boundary of the coating, Chen et al. (Arch. Ration. Mech. Anal. 206 (2012) 911-951) derived the complete list of effective boundary conditions satisfied by the limiting model. In this paper we provide explicit error estimates between the full model and the effective model. Moreover, our error estimates are independent of time, which shows that the maximal time interval in which the effective boundary conditions remain valid are infinite. The proof is based on

H^{2}

estimates for solutions of the full model, characterization of large time behaviors for solutions of the effective model, and interaction estimates between the two models.

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最佳排列涂层热方程有效边界条件的误差估计

我们感兴趣的是，当涂层厚度收缩为零时，涂层体上热方程的有效边界条件是否有效。涂层是最佳排列的，即涂层中的法向量是热张量的特征向量。如果热方程满足涂层外部边界的诺伊曼边界条件，陈等人（Arch.Ration.力学。Anal.206 (2012) 911-951）推导出了极限模型所满足的有效边界条件的完整列表。在本文中，我们提供了完整模型与有效模型之间的明确误差估计。此外，我们的误差估计与时间无关，这表明有效边界条件保持有效的最大时间间隔是无限的。证明基于完整模型解的 H2 估计、有效模型解的大时间行为特征以及两个模型之间的交互估计。

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

Journal of Mathematical Analysis and Applications 数学-数学

CiteScore

2.50

自引率

7.70%

发文量

790

审稿时长

6 months

期刊介绍： The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions. Papers are sought which employ one or more of the following areas of classical analysis: • Analytic number theory • Functional analysis and operator theory • Real and harmonic analysis • Complex analysis • Numerical analysis • Applied mathematics • Partial differential equations • Dynamical systems • Control and Optimization • Probability • Mathematical biology • Combinatorics • Mathematical physics.