PhD

I am always happy to supervise PhD. If you are interested in doing PhD with me, please contact me and click here to check your eligibility for Monash PhD admission and scholarship. Currently, the topics for PhD include:

Harmonic analysis: to study the estimates in various function spaces for the linear/multilinear operators related to the Fourier restriction problem and PDEs, function spaces.

Nonlinear partial differential equation: to study the nonlinear evolutionary PDEs arising in mathematical physics, main concerns are the low-regularity local/global well-posedness theory, long-time/blow-up behavior of the solution, stability of the equation, playing tools from many other areas including harmonic analysis, functional analysis, ODE and probability, etc.

Honours projects

1. Fourier restriction estimates Fourier restriction estimate is one of the core topics in harmonic analysis. In 1960s, E. M. Stein observed for the first time the restriction phenomenon of the Fourier transform, and proposed the Fourier restriction conjecture. This conjecture in 3 and higher dimensions is still open and has been studied extensively. Besides revealing a fundamental property of the Fourier transform, Fourier restriction estimate has found tremendous applications in other fields, e.g. PDEs. This project will be devoted to study the classical Fourier restriction results and its applications in PDEs, providing a friendly introduction to the fascinating world of analysis and PDEs.

2. Lp eigenfunction estimates on compact manifold This project will be devoted to the study of Lp estimates for the eigenfunction of the Laplacian operator on compact manifold. These estimates are fundamental and have been the one of main topics in the field of analysis and geometry. Through this project, you will play with tools from analysis, geometry and dynamical system.

3. 3D axisymmetric Navier-Stokes equation The global existence of the large smooth solution for the 3D Navier-Stokes equation is a long-standing open problem. It is one of the millennium-problems. This project will focus on the axisymmetric case. This case was known to have some more structures so that there are rich results. Especially recently the criticality for this special case was revealed by Lei-Zhang basing on the work of Chen-Fang-Zhang. Their results showed it is only logarithmically far to finally solve the problem for this case. This project will be devoted to study the classical results, the rich structure as well as the recent mentioned works

4. Some topics in nonlinear dispersive equations In the last 30 years, there were enormous developments in the fields of nonlinear dispersive equations, e.g. KdV, Schrodinger and wave equations. Many new tools were developed and new ideas from other fields have played important roles. This project will focus on one of the following topics: low regularity well-posedness problems, long-time behavior, blowup behavior, and probabilistic well-posedness.

5. Probabilistic methods in analysis The probabilistic methods have been applied in many problems in analysis and PDEs, and have proved to be extremely powerful. In particular, they are useful, such as in showing the existence of some counter examples when constructive method is difficult to formulate, and in providing new perspectives and directions when the deterministic problems are sometimes difficult to study. This project will address the recent applications of probabilistic methods to some problems in modern analysis and PDEs.

Summer research projects

Want to have a taste on the current research beyond your textbook? Apply the summer research fund from AMSI or Monash. Projects 1, 3 and 5 in above Honours projects are also for this summer program.