The main focus of my talk shall be on the well-posedness for the interface problem between a viscous fluid and an elastic solid. This is a two-phases problem, where each phase satisfies its own natural equation of evolution, and where the interaction between the two phases comes from the natural continuity of velocity field and normal stress across the unknown moving
interface. The methods known in fluid moving boundary problems (viscous or inviscid) cannot handle the apparent incompatibility between the regularity of the two phases, which has led previous authors to consider the case where the solid satisfies a simplified law where the difficulties are not present. I shall present the new methods that where required in order to allow the treatment of classical elasticity laws in this moving
interface problem.

I shall then briefly explain how some of these ideas and some new tools preserving the transport structure of the Euler equations can provide the well-posedness for the free surface Euler equations with (or without) surface tension, without any restriction on the curl of the initial velocity.