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Selected Development Project
 
Project Title

Wellposedness and Vanishing Diffusion Parameters Limits of the Magneto-geostrophic Equation
磁地轉方程的適定性及消失的擴散參數極限

 
Principal Investigator Dr SUEN Chun Kit Anthony
 
Area of Research Project
Mathematics and Information Technology
 
Project Period
From 01/2017 To 12/2019
Objectives
  1. To obtain global-in-time wellposedness of the three-dimensional magneto-geostrophic (MG) equation for different cases depending on the diffusion parameters with the presence of damping and forcing terms.
  2. To address the convergence problems of solutions of the three-dimensional magneto- geostrophic (MG) equation in the vanishing diffusion parameters limits.
  
Methods Used

The main principle behind this project is to exploit the nature of the operator Mυ that produces the drift velocity u from the scalar field ϑ present in the MG Equation. The operator Mυ produces two orders of smoothing when υ>0, which allows us to prove global-in-time wellposedness of the MG equation. On the other hand, we address carefully the behaviour of Mυ as υ tends to zero and the relation between υ and the viscosity κ. To this end, we expect to use advanced harmonic analysis theories and discover new methods for proving existence, uniqueness, long-time behaviour and regularity properties to certain system of partial differential equations.
Summary of Findings

We successfully prove the global existence of classical solutions to a class of forced drift-diffusion equations with L2 initial data and divergence free drift velocity uυ in Lt∞BMOx-1, and we obtain strong convergence of solutions as υ vanishes. We then apply our results to the three dimensional magneto-geostrophic (MG) equation as described in this project. We further discuss the dissipation of energy and the existence of a compact global attractor in L2 for the critical MG equation.
Impact

New mathematical methods for proving existence, uniqueness and regularity will be developed in this project which are also applicable to the study of other types of active scalar equations including surface quasi-geotropic equation (SQG) and Euler equation. New Fourier analysis methods will also be needed for the study of abstract functional operators. The overall activity described in the project will stimulate new collaborations between applied mathematics and geophysics while original articles will be published in international peer-reviewed journals to spread the new results.
Selected Output
S. Friedlander and A. Suen, Solutions to a class of forced drift-diffusion equations with applications to the magneto-geostrophic equations. Annals of PDE, 4(2):14, Sep 2018.
Biography of Principal Investigator

Dr Suen Chun Kit Anthony is an Assistant Professor at the Department of Mathematics and Information Technology at The Education University of Hong Kong. His research focuses primarily on the existence theory and large-time behavior of solutions of certain systems of partial differential equations. In particular, he is interested in the Navier-Stokes equations of multidimensional, compressible/incompressible fluid flow with applications to magnetohydrodynamics (MHD), magnetogeostrophic equation (MG) and viscoelastic flow. He also studies mathematical biology such as chemotaxis and blood flow modeling.
Funding Source

Early Career Scheme