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.

 In this talk, I will discuss recent results on the
 large time well-posedness of classical solutions to the
 multi-dimensional compressible Navier-Stokes system with possible
 large oscillations and vacuum.
 The focus will be on finite-time blow-up of classical solutions for
 the 3-D full compressible Navier-Stokes system, and the global
 existence of classical solutions to the isentropic compressible
 Navier-Stokes system in both 2-D and 3-D in the presence of vacuum
 and possible large oscillations.  New estimates on the fast decay
 of the pressure in the presence of vacuum will be presented,  which
 are crucial for the well-posedness theory in 2-dimensional case.

In recent work with Guy David we introduce the notion of almost
minimizer for a series of functionals previously studied by Alt-Caffarelli
and Alt-Caffarelli-Friedman.

We prove regularity results for these almost minimizers and explore the
structure of the corresponding free boundary. A key ingredient in the study
of the 2-phase problem is the existence of almost monotone quantities. The
goal of this talk is to present these results in a self-contained manner,
emphasizing both the similarities and differences between minimizers and
almost minimizers.

We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about recent results on existence of regular reflection solutions for potential flow equation up to the detachment angle, and discuss some techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear equation of mixed elliptic-hyperbolic type. Open problems will also be discussed. The talk is based on joint work with Gui-Qiang Chen.