Mathematical analysis for local structures of viscous incompressible flows

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It is well-known that vorticity fields play important roles in dynamics of incompressible flows. This research aims to analyze linear and nonlinear partial differential equations related with vorticity fields mathematically. The research in particular has made important contributions in the following topics:(i) Stability analysis of some stationary solutions modeling vortex tubes in turbulent flows;(ii) Estimates of fundamental solutions to fractional diffusion equations with a drift;(iii) Analysis of vorticity equations in the half plane and its applications to inviscid limit problem for the Navier-Stokes equations.