My main theme is to construct a mathematical theory for nonlinear phenomena, based upon Partial Differential Equations. Some of the concrete major subjects are:
Initial-boundary value problems, Free boundary problems, Initial-value problems with general Navier's slip conditions and Boundary-value problems in angle domains for compressible and incompressible Navier-Stokes equations; Free boundary problems for incompressible Euler equations; Water waves; Vortex filament equation and vortical phenomena; Stefan problems and analysis of phase transition and critical phenomena; (visco-) elastic equations in cracked domains; nonlinear acoustic equations; Primitive equations; non-Newtonian equations; Inverse problems etc.