Interviews with Faculty Researchers

– Interview with Prof. Hu Xianpeng
Professor, Department of Applied Mathematics

In the realm of mathematical analysis pertaining to mechanics models, Prof. Hu Xianpeng has made important contributions, particularly through his investigations into Partial Differential Equations (PDEs). His research has made significant contributions to understanding the well-posedness and regularity aspects inherent in models within mechanical areas. Given the inherent complexity of mechanical systems, Prof. Hu's work has been pivotal in elucidating fundamental mathematical principles within this domain.

Central to the mathematical analysis of mechanics models are concepts like well-posedness, regularity, and the distinction between weak and strong solutions, which ensure the stability, predictability, and smoothness or differentiability of mathematical models, respectively. Weak solutions, though less stringent, offer valuable insights into systems where classical solutions might be elusive, presenting one of the primary challenges in the mathematical theories of PDEs. Prof. Hu highlights the necessity of relaxing initial conditions to facilitate a more tractable analysis, advocating for the consideration of weaker solutions initially and subsequent verification of differentiability, enabling researchers to navigate complex systems more effectively.

Moreover, Prof. Hu emphasizes the complementary nature of weak and strong solutions. While weak solutions provide a pragmatic framework for understanding physical phenomena, strong solutions offer a foundation for rigorous mathematical analysis. Striking a balance between these concepts is essential for a comprehensive understanding of mechanics models.

Prof. Hu's work not only exemplifies the interdisciplinary nature of mathematical analysis in mechanics but also underscores the symbiotic relationship between mathematics and physics. By elucidating complex physical phenomena through mathematical theories, his research enables researchers to refine models, design algorithms, and make predictions with greater accuracy, thus enriching theoretical mathematics and enhancing the practical applicability of mechanics models in engineering and science. 

經典力學模型的偏微分方程進展

– 胡先鵬教授專訪
應用數學系教授

胡先鵬教授的研究對力學模型的相關數學分析作出了很大貢獻，當中以其偏微分方程（PDE）的研究尤為突出。在理解力學模型固有的適定性和正則性方面，鑑於力學模型有其固有複雜性，難以從基本數學原理來清楚闡明，胡教授的研究對此作出了清晰的論述，而他的研究在這方面亦發揮了相當關鍵的作用。

他認為力學模型數學分析的核心概念是適定性、正則性以及弱解和強解之間的分別，它們分別奠定了相關數學模型的穩定性、可預測性、光滑性或可微分性。弱解雖然不太嚴格，但卻為傳統解决方案難以解釋的系統提供參考價值，這亦是 PDE 數學理論的主要挑戰之一。胡教授指出了寬鬆的首要條件對分析十分重要，並表示首先考慮弱解，然後驗證可微分性，可讓研究人員能更靈活地駕馭複雜的系統。

此外，胡教授亦強調弱解和強解具有互補的特性。弱解為理解物理現象提供了一個實用框架，而強解則為嚴謹的數學分析奠下基礎。對全面分析力學模型而言，能否在這些概念之間取得平衡十分重要。

胡教授的研究工作不僅反映了數學分析與力學兩者具有跨學科的關連性質，亦突顯了數學與物理學之間的互動關係。在過去，他的研究一直利用數學理論來闡明複雜的物理現象，讓研究人員能夠完善模型、設計演算法，並更準確地進行預測，從而為數學帶來更豐富的理論，以及讓力學模型在工程和科學中的實際應用更加廣泛。