Internal waves are traveling waves that propagate along the interface dividing two immiscible fluids. In this talk, we discuss recent progress on rigorously constructing two related species of extreme internal waves: overturning bores and gravity currents. Extreme refers to the fact that there is a stagnation point on the interface, which allows for the boundary to be non-smooth. 

Hydrodynamic bores are front-type traveling wave solutions to the two-layer free boundary Euler equations in two dimensions. We  prove that there exists a family of solutions of this form that starts at  trivial solution where the interface is flat and continues until the interface develops a vertical tangent. This type of behavior was first observed over 40 years ago in computations of internal gravity waves and gravity water waves with vorticity via numerical continuation. Despite considerable progress over the past decade in constructing global families of water waves that potentially overturn, a rigorous proof that overturning definitively occurs has been stubbornly elusive.  

Gravity currents arise when a heavier fluid intrudes into a region of lighter fluid. Typical examples are  muddy water flowing into a cleaner body of water and haboobs (dust storms). We give the first rigorous proof of a conjecture of von Kármán on the structure of gravity currents near the rigid boundary.