Free interface wave motions in fluid dynamics and stabilizing effects of magnetic fields

Interfacial wave motions, such as shock waves, water waves, vortex sheets, jet flows, and multi-phase flows etc., are fundamental flow patterns in fluid dynamics and give rise to many important free boundary problems in various PDEs. The well-posedness and stability (both structural and dynamical) of such wave motions have been not only a core of theoretical research, but also vast applications in physics and engineering. Despite the tremendous important progress made in the past, there are still many outstanding challenges in the area.

In this talk, we first give a rough introduction of some typical free interface motions, then survey some progress, and indicate some open problems. In particular, as two examples, we will discuss the stabilizing effects of the transversal magnetic fields on the motions of MHD entropy waves and global existence of plasma-vacuum and plasma-plasma interfaces. The emphasis will be on outlining some unsolved problems.