Inviscid active scalar equations belong to a special type of transport equations in which the scalar function is related to the velocity field via some constitutive laws. 

This kind of differential equations comes from numerous physical models such as the Navier-Stokes, Euler, or magneto-hydrodynamic equations, all of them have great importance in practical applications ranging from fluid mechanics to atmospheric science, oceanography and geophysics. 

In this research project, the PI proposes to study different mathematical questions relating to well-posedness and possible formation of finite time singularity of a general class of inviscid active scalar equations, which arises in a variety of physical models including the surface quasi-geostrophic equation, incompressible porous media equation and inviscid magneto-geostrophic equation. 

The primary goal of this project is to achieve some fundamental results on a general class of inviscid active scalar equations in various regimes of parameters together with some specific constitutive laws that relate a scalar function with the velocity field. Such proposed problem is difficult mainly due to the fact that the constitutive laws will behave drastically different in varying regimes of parameters, and the success of this problem will impact the underlying physical science, in the way that by establishing well-posedness and discovering possible singularity for these mathematical models, it can help us explain and recognise the corresponding physical phenomena at a more profound level. 

The secondary goal of this project is to apply the results from the general classes of inviscid active scalar equations to some well-known physical models with emphases on diversified aspects. Such problem is crucial but challenging because these classical systems in fluid mechanics or physical science are not yet fully understood. 

The success of this problem will bring us some brand-new ways in studying these physical phenomena, which can be achieved by obtaining well-posedness and identifying singularity formation. In the short run, this project can initiate and stimulate collaborations between Hong Kong scholars and other international experts, which are beneficial for future cooperations between various parties beyond academia. 

Furthermore, this project can provide chances for undergraduate and graduate students to study various theoretical area of applied analysis, which helps them get prepared for later research. In the long run, this project will eventually bring good impacts on our society in the sense that it enriches our understanding on different branches of physical science, which can help improve our living environment and quality of life.