Interviews with Faculty Researchers

– Interview with Prof. Yang Tong
Chair Professor of Mathematical Science, Department of Applied Mathematics

The study of the motion of fluids, such as liquids and gases, has a long history and has been facing challenges posed by irregular motion. Prof. Yang Tong, Chair Professor of Mathematical Science of the Department of Applied Mathematics said that the mathematical theories of gas motion can offer valuable insights into the fundamental phenomena and applications in various scientific and engineering disciplines, such as aerodynamics and hydrodynamics.

Instead of looking into the precise physical movement of each particle, kinetic theory provides an approach to investigate how the distribution function of particles evolves in time so that the macroscopic quantities, such as density, velocity, and temperature can be evaluated against physical observations.

Different mathematical theories can be applied to study the motion of different fluids, depending on the physical situation in which different parameters and scaling are taken into account. For example, macroscopic models like Euler and Navier-Stokes equations model water movement, while the Boltzmann equation models rarefied gas.

In addition to atmospheric pressure and airflow, external factors such as the Lorentz force induced by electromagnetic fields, and the force induced by boundary effects, are also considered when studying gas dynamics. While general attempts are made in mathematical theory, some conditions are often imposed to achieve well-posedness theory in a suitable function space.

Despite the breadth of the study of gas dynamics, numerous significant challenges remain, including Hilbert's sixth problem about Boltzmann equation, and the Millennium problem on Navier-Stokes equations. Probability is crucial to understanding gas dynamics as the Boltzmann equation is derived from statistical physics. While some exemplary works utilizing probability approaches to study gas dynamics exist, more research is necessary to achieve a complete understanding of these systems. Ultimately, developing mathematical models of gas motion necessitates a profound understanding of physical phenomena and rigorous mathematical analysis.

As Prof. Yang believes, one of the main objectives of investigating gas motion using mathematical theories is to inspire students and scholars to pursue research in theoretical mathematics, which can bring joy in problem-solving, and satisfaction in revealing the elegance of concepts and theories for practical applications. Additionally, developing a stable and promising academic career for young scholars in a stimulating and supportive environment is equally important.

氣體運動之數學理論

– 楊彤教授專訪
應用數學系數學科學講座教授

包括液體和氣體在內的流體動力研究歷史悠久，卻一直面臨流體不規則運動的挑戰。應用數學系的數學科學講座教授楊彤教授認為，相關數學理論在解釋空氣動力學、流體動力學等各範疇的基本現象和應用時，能提供寶貴的參考價值。

有別於精確地研究每個粒子的物理運動規律，動理學理論提供了相關分布函數如何隨時間演變的方法，以對密度、速度和溫度等宏觀變量進行觀察和評估。不同的數學理論可以應用於研究各類型流體的運動，但當中的參數和測量準則，則取決於實際環境因素。例如，歐拉和納維爾－斯托克斯方程式等可以用來解釋水的運動規律；玻爾茨曼方程式則可以用來解釋稀薄氣體的運動規律。

另外，研究還需要考慮到外部因素，如電磁場引起的洛倫茲力、以及邊界效應引發的力。雖然一般都想將一套理論應用於所有情況，但建立時往往需要考慮一些相應條件，才能在合適的函數空間裡建立嚴格的論。

氣體動力學的研究範圍十分廣泛，但仍存在大量挑戰，包括玻爾茨曼方程式的希爾伯特第六問題，以及納維爾-斯托克斯方程式的「千禧問題」。概率對於理解氣體動力學而言極其重要，因為玻爾茨曼方程式是從統計物理學中推導出來的。雖然現時已有利用概率研究氣體動力學的成功例子，不過還需要更深入的研究，才能掌握當中的理論。如果最終想建立新的數學模型，我們更需要對物理現象具備深刻的理解和進行嚴謹的數學分析。

楊教授認為，相關研究可激發學生和學者積極探討數學理論之美。當中不僅令人在解難的過程中感到快樂，還能揭示概念和理論在實際應用方面的巧妙，以帶來滿足感。為年輕學者提供具啟發性的研究環境和充份支援同樣重要，這讓他們的學術生涯能夠穩定發展及帶來前景。