Understanding the incompressible/compressible fluid is a fundamental, but
challenging, project not only in numerical analysis, but also in
theoretical analysis, especially when extra effects, such as the elastic
deformation or the magnetic field, interact with the flow. In this talk,
the incompressible fluid and its associated flow map will be reviewed first.
The main object of this talk devotes to a recent work in understanding
incompressible/compressible magnetohydrodynamic fluids with zero magnetic
diffusivity (which is equivalent to infinite conductivity). This is a
joint work with Fanghua Lin.