On the Fritz John problem for a freely floating object

The Fritz John problem is a linear description of the interactions of an incompressible irrotational free surface fluid with a partially immersed solid object. The model was introduced by John in 1949 and is an important model closely linked to the Cummins equations, which are widely used by naval engineers to compute wave-structure interactions. However, despite their active use, it remains an open question whether the Cummins equations are well-posed. The main difficulty arises from the presence of corners in the fluid domain, which leads to the formation of singularities. In this talk, I will present recent work in collaboration with David Lannes that addresses the well-posedness issue. Moreover, we study the singularities at the corners and their impact on the regularity of the solution.