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

Nonlinear partial differential equation: to study the nonlinear evolutionary PDEs arising in mathematical physics such as nonlinear dispersive equations, Navier-Stokes equation.  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.

Harmonic analysis and PDEs