Bring us some brand-new ways in studying magnetic relaxation, which can be achieved by studying the long time behaviour of solutions with the help of the notion of vanishing viscosity weak solutions. 

Active vector equations can be regarded as a special type of differential equations in which the unknown vector-valued function is related to the velocity field via some constitutive laws. This kind of differential equations comes from many physical models such as the incompressible Euler equations or magnetohydrodynamics (MHD), all of them have great importance in practical applications ranging from fluid mechanics to geophysics, astrophysics, cosmology and engineering. 

The primary goal of this project is to achieve some fundamental results on a general class of viscous active vector equations in various regimes of parameters, which include the globalin-time well-posedness and the convergence of solutions when the diffusive parameters vanish. Such proposed problem is fruitful in the sense that by establishing well-posedness under the smoothing effects on the magnetic and velocity vector fields, it provides approximate solutions for the related inviscid models that will help identify singular behaviours as those diffusive parameters vanish. 

The secondary goal of this project is to rigorously address a class of inviscid active vector equations arising from the viscous models that are just mentioned in the primary goal. Such problem is crucial but challenging because these inviscid models come from a topology preserving diffusion process known as magnetic relaxation, in which its physical and mathematical properties are not yet fully understood. The success of this problem will bring us some brand-new ways in studying magnetic relaxation, which can be achieved by studying the long time behaviour of solutions with the help of the notion of vanishing viscosity weak solutions. 

In the short run, this project forms collaborations between Hong Kong scholars and international experts around the world, which are important for further cooperations between various parties beyond academia. Furthermore, this project can provide chances for undergraduate and graduate students to dive in some new theoretical areas of applied mathematics, which help them get prepared for their future research career. In the long run, this project will eventually bring good impacts on our society in the sense that it enriches our understanding on fluid dynamics and its related branches of physical science, which can help unveil the mystery of the Earth as well as our universe.