Séminaire Analyse non linéaire et EDP

The streamlines of periodic irrotationaltraveling water waves are known to be real-analytic,with exception of the free surface in the case thewave of greatest height which has a corner at the wavecrest (the lateral tangents being at an angle of 2pi/3).The regularity of waves of small and moderate amplitudeis, perhaps surprisingly, little affected by thepresence of vorticity in the flow. This is joint workwith J. Escher.

The lubrication approximation leads to a fourth order degenerate equation modeling the evolution of small viscous droplets on a solid support (the thin film equation). Along the contact line (the free boundary), the solution must satisfy a gradient condition (contact angle condition).While many existence and regularity results are known for solutions with zero contact angle, the only existence result with non-zero contact angle is due to F. Otto and only holds in some particular framework (Hele-Shaw cell). We consider a singular perturbation approach to generalize Otto's result.