In this research project, the PI proposes to deepen our understanding of active scalar equations with emphasis on the vanishing viscosity limit and long time behaviour of a family of active scalar equations, as a key step in studying some physical models which include the magneto-geostrophic equation and incompressible porous media Brinkman equation. 

Active scalar equations are well-known and important topics in mathematics. They have numerous physical applications which include fluid mechanics, magnetohydrodynamics, atmospheric science and geophysics. Tremendous progress has been made by various mathematicians on the analytical theories of active scalar equations, yet due to the nonlinear nature of the equations, many problems remain as open and challenging. In this research project, the PI proposes to deepen our understanding of active scalar equations with emphasis on the vanishing viscosity limit and long time behaviour of a family of active scalar equations, as a key step in studying some physical models which include the magneto-geostrophic equation and incompressible porous media Brinkman equation. The first topic in this proposal focuses on different scenarios when some viscosity type parameters tend to zero in the finite or long time regimes. The modes of convergence of solutions, existence of certain types of attractor as well as the associated properties will be carefully addressed. Such proposed problem is difficult because the long time inviscid limits of many forced systems in fluid mechanics are not fully understood, and the success of this problem will shed lights on understanding some ideally conserved quantities such as kinetic energy and enstrophy. The second topic in this proposal applies the results from the general classes of active scalar equations to different physical models including magneto-geostrophic equation and incompressible porous media Brinkman equation. Such problem is challenging due to the highly singular limits of different vanishing viscosities and the drastic changes in the correlation between velocity field u and the corresponding scalar function θ. The success of this problem will significantly extend and strengthen the results for magneto-geostrophic equation and incompressible porous media equation from the literature. In the long run, this project may lead to discoveries of new techniques from the fields of analysis and partial differential equations which enrich our knowledge about fluid mechanics. On the other hand, our project can attract undergraduate and graduate students to such theoretical area of applied analysis and nurture their future developments.