发表机构
University of Pavia; Consiglio Nazionale delle Ricerche; Università degli Studi di Parma（帕维亚大学; 意大利国家研究委员会; 帕尔马大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对薄膜-体域热传导问题，通过缩放假设推导薄膜极限的耦合界面公式，建立适定性并给出显式收敛速率，辅以解析解验证。
AI 中文摘要
我们研究了一个由薄膜区域和体域区域组成的域中的热传导问题，这两个区域通过一个界面耦合，在该界面上施加了温度连续性和热通量平衡条件。分析涵盖了有限厚度薄膜构型以及相应的薄膜极限，其中热导率张量是各向异性且非均匀的。直接令薄膜厚度趋于零会导致一个退化问题，因为薄膜区域的测度趋于零。为了在极限中保留薄膜的贡献，我们引入了对法向热通量的适当缩放假设，即针对薄膜-体域界面上的通量平衡以及施加在薄膜区域外部边界上的通量，同时要求薄膜中的体积源相对于薄膜厚度一致有界。在这些缩放假设下，我们推导出一个非平凡的薄膜极限公式，其形式为耦合的体域-界面问题，其中塌缩薄膜的影响通过一个附加的界面平衡关系来表示。对于有限厚度薄膜和薄膜极限构型，都建立了适定性结果，并进行了严格的收敛性分析以证明渐近极限的合理性。主要的数学成就在于在合适的函数范数中找到了显式的收敛速率，这仅依赖于有限厚度问题由于薄膜界面处的透射条件而具有的弱正则性。为了补充这项研究，我们为一个简化的薄膜-体域几何结构推导了解析解，并用于比较两种构型的热响应。这些解还提供了对理论分析中建立的收敛速率的定量评估。
英文摘要
We investigate a thermal transmission problem in a domain composed of a film region and a bulk region coupled through an interface where temperature continuity and heat-flux balance are imposed. The analysis addresses both the finite-thickness film configuration and the corresponding thin-film limit, with anisotropic, non-homogeneous thermal conductivity tensors. Letting the film thickness vanish directly leads to a degenerate problem due to the vanishing measure of the film region. To preserve the contribution of the film in the limit, we introduce suitable scaling assumptions for the normal heat fluxes, namely for the flux balance across the film-bulk interface and for the flux prescribed on the external boundary of the film region, together with the requirement that the volumetric source in the film be uniformly bounded with respect to the film thickness. Under these scaling assumptions, we derive a non-trivial thin-film limit formulation that takes the form of a coupled bulk-interface problem, where the effect of the collapsing film is represented through an additional balance relation posed on the interface. Well-posedness results are established for both the finite-thickness film and thin-film limit configurations, and a rigorous convergence analysis is carried out to justify the asymptotic limit. The main mathematical achievement consists in finding explicit convergence rates in suitable functional norms, relying only on the weak regularity of the finite-thickness problem due to the transmission conditions at the film interface. To complement the study, analytical solutions are derived for a simplified film-bulk geometry and are used to compare the thermal responses of the two configurations. These solutions also provide a quantitative assessment of the convergence rates established in the theoretical analysis.
CommentsThe present work is dedicated to the memory of our dear friend and esteemed colleague Prof. Alessandro Veneziani, recalling his passionate enthusiasm and pioneering intuition in combining mathematics, engineering, and physics, as well as his constant dedication to collaboration and mentoring. 46 pages, 7 figures
