Large-amplitude internal waves in stratified fluids exhibit striking nonlinear phenomena, including overturning fronts and gravity currents. In this setting, extreme refers to traveling waves that develop a stagnation point along the interface, allowing the free boundary to lose regularity and form singular features such as vertical tangents. Although numerical continuation studies dating back more than forty years strongly suggest that such behavior occurs for solutions of the two-layer free-boundary Euler equations, a definitive mathematical proof has remained open.

In this talk, I will describe the construction of a global family of hydrodynamic bores, which are front-type traveling waves connecting distinct asymptotic states, bifurcating from the trivial flat interface. I will show that along the elevation branch the waves must overturn, and the interface necessarily develops a vertical tangent. This yields the first rigorous proof of overturning obtained through a global bifurcation framework in the fully nonlinear O(1) gravity regime.

Along the depression branch, the limiting configuration instead produces a gravity current, a physically fundamental flow in which a denser fluid intrudes beneath a lighter one. I will also explain how this analysis leads to a proof of a classical conjecture of von Kármán describing the structure of gravity currents near a rigid boundary.

This is a joint work with Samuel Walsh (Missouri) and Miles Wheeler (Bath).