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

• Published in: Journal of mathematical analysis and applications Vol. 543; no. 1; p. 128972
• Main Authors: Meng, Lixin, Zhou, Zhitong
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2025
• Subjects: Effective boundary conditions; Eigenvalue; Error estimates; Heat equation; Optimally aligned coatings; Eigenvalue; Error estimates; Heat equation; Effective boundary conditions; Optimally aligned coatings
• ISSN: 0022-247X
• DOI: 10.1016/j.jmaa.2024.128972

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 H2 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.