Four outstanding scholars from City University of Hong Kong (CityU) have been selected as awardees of the inaugural Senior Research Fellow Scheme (SRFS) and Research Fellow Scheme (RFS) under the Research Grants Council (RGC), being granted a sum of more than HK$23 million as research funding. The acknowledgement of their research achievements is testimony to CityU’s research strengths and recognition of the University’s efforts in fostering interdisciplinary research.
Professor Tong Yang : partial differential equations and kinetic theories
Professor Tong Yang, Chair Professor of Mathematics, was granted the title “RGC Senior Research Fellow”. He has been working on problems related to partial differential equations and kinetic theories. He will use the SRFS grant to develop analytic techniques that can be applied to studies on solution behaviour and fluid dynamic limits of some typical kinetic models.
The study on the gas motion and fluid dynamics can be traced back to the 17th century when Sir Isaac Newton quantified the fluid phenomena by Newton equations (Newton's law of viscosity). Tremendous progress was made from the 18th to the 20th century when many famous equations were formulated, such as the Bernoulli’s law, Euler equation, Navier-Stokes equation, and Prandtl equation.
Many famous equations of motion have been derived by focusing on different aspects of gases and fluids in different physical scales. In the macroscopic scale where the gas and fluid are regarded as a continuum, their motion is described by the macroscopic quantities such as macroscopic mass density, temperature, and pressure. In this scale, the Euler and Navier-Stokes equations are the most famous equations among the governing systems proposed in fluid dynamics. And these two fundamental systems of equations are typical examples of the conservation laws.
In opposite to the macroscopic scale, gas and fluid are viewed as a many-body system of microscopic particles, such as atoms and molecules, in the microscopic scale. The motion of the system is governed by the coupled Newton equations, within the framework of classical mechanics. Since the movement of each atom takes place in the three-dimensional space, an enormous number of Newton equations would be involved in the calculation. As a result, it is not practical to use the Newton equations because solving such a large coupled system and specifying all the initial data are impossible. Statistics and probability are to be used for calculation instead. On the other hand, the macroscopic fluid dynamical quantities mentioned above are related to the statistical average of the quantities depending on the microscopic state.
The kinetic theory gives the mesoscopic (between the macroscopic and microscopic scales) description of the movement of gas and fluid. It is the fundamental theory that links the theories of microscopic and macroscopic scales. And the Boltzmann equation is the most fundamental equation in the kinetic theory.
In this project, Professor Yang aims at studying the solution behaviour and fluid dynamic limits of some typical kinetic models, including the Vlasov-Maxwell-Boltzmann system and the Vlasov-Nordstr?m-Fokker-Planck system. The analytic techniques developed in this project may be applicable to the study on other systems of kinetic equations, so as to enrich the existing mathematical theories in this important area.
This research article originated from CityU Research Stories.