Active scalar equations are ubiquitous in nature, science and engineering. They are of great importance in practical applications ranging from fluid mechanics to atmospheric science, oceanography and geophysics. Due to the nonlinear and nonlocal nature of the equations, many problems on active scalar equations remain as open and challenging. 

In this research project, the PI proposes to address various mathematical questions relating to wellposedness, long time behaviour and anomalous dissipation of a general class of active scalar equations, which arises in a variety of physical models including the magneto-geostrophic equation, incompressible porous media Brinkman equation and surface quasi-geostrophic equation. The primary goal of this project is to achieve some fundamental results on a general class of active scalar equations in different regimes of diffusive parameters. Such proposed problem is difficult because of the intricate interplay between different diffusive parameters, and the success of this problem will impact the underlying physical science, in the way that by insuring well-posedness of these mathematical models, it can help to understand and validate the corresponding physical phenomena at a deeper level. The secondary goal of this project is to apply the results from the general classes of active scalar equations to some physical models, with emphases on different aspects which are tailor-made for these physical models. Such problem is challenging and important because the singular inviscid limits of many forced systems in fluid mechanics or physical science are not fully understood, and the success of this problem will provide us a more effective and realistic way in studying these physical phenomena. In the short run, this project can lead to scientific results which will be published on international high-standard journals, and it will certainly stimulate collaborations between Hong Kong scholars and other international experts which can enhance further cooperations between various parties. Furthermore, this project can attract undergraduate and graduate students to such theoretical area of applied analysis and nurture their future developments. In the long run, this project will ultimately bring benefits to our society in the sense that it enriches our understanding on other branches of physical science, which can be both beneficial and crucial for improving our quality of life.