Supervision

I am happy to supervise students at different levels.  Please read below for more descriptions of my field.  If you are interested, please feel free to contact me and introduce yourself with 1) CV; 2) Transcripts; 3) English results (optional); 4) Other documents (optional), e.g., publications, certificate.

• Dispersive PDE

The terminology “dispersive” is from optics.   The well-known dispersion phenomenon in optics is caused by the fact that in the prism lights with different colours (frequency) have different propagation speeds.   In mathematical language, the phase velocity of a wave depends on its frequency.  See here for more explanation.

Dispersive PDE forms an important class of evolution equations (the unknown functions depend on time variable t and space variables x) which arise in mathematical physics, geometry and other fields.  It studies the propagation of waves, but with a variable propagation speed depending on the frequency of the wave.  Dispersive PDE is different from the traditional classes (e.g. elliptic, parabolic and hyperbolic types). It has some similar properties to the hyperbolic type such as wave equations which have constant propagation speed.  It contains a large class of Hamiltonian PDE, such as the Schrodinger equation in quantum mechanics, and the Korteweg-de Vries equation in fluid mechanics. 

For a PDE, a fundamental task is to understand its solutions.  We study the qualitative and quantitative properties of the solutions through strict analysis without solving the equation explicitly. 

• Low regularity well-posedness
  existence, uniqueness and continuous dependence

• Global in time well-posedness VS blow up in finite time

• Long-time dynamics for global solutions

• Formation of singularities for blow-up solutions

• Special solutions, e.g., solitary solutions, self-similar solutions and their properties

• Global perspective as a dynamical system, instead of individual solutions

• Stochastic aspects of dispersive equations

• Numerical methods

• Harmonic analysis

The central roles in harmonic analysis are various operators that have general backgrounds and motivations from analysis, PDE, and other fields.   The purpose is to study the behaviour of these operators, e.g. boundedness, limiting behaviour.  Usually,  in the study one applies the "divide and conquer" strategy.  First divide the operators into sub-pieces according to some patterns, then conquer each sub-piece using different strategies.  A typical tool is the "Whitney decomposition" performed to the prism.  Tools like this help to understand the dispersion phenomena.  Harmonic analysis also provides many tools for "hard analysis" of many problems, where some subtle structures play crucial roles and need to be exploited: 

• Cancellation
  singular integral operator, commutator, compensated compactness (NULL structure)

• Orthogonality
  almost orthogonality, square root cancellation, decoupling inequality (orthogonality in the non-Hilbert space, e.g.  Lp space)

• Oscillation
  Oscillatory integral operator

• Function spaces
  Hardy space, BMO space, adaptive function spaces associated to the operators.

• Interpolation techniques

In particular, the solution map of a nonlinear PDE is a nonlinear operator.  Harmonic analysis tools naturally play crucial roles in studying it, e.g., a core task is to design some adaptive function spaces to the PDE in consideration.