While the dynamics of compressible fluids and shock waves form the core of this conference, this talk explores an analogous but distinct class of coherent structures in incompressible stratified fluids: large-amplitude internal waves that develop singularities at a free interface. These “extreme” waves are characterized by the formation of a stagnation point, leading to loss of regularity, including vertical tangents and overturning fronts, reminiscent of limiting configurations in hydraulic jumps or bores.

Despite strong numerical evidence spanning four decades that such singular behavior occurs in solutions of the two-layer free-boundary Euler equations, a rigorous mathematical proof has remained elusive. In this talk, I will describe the construction of a global family of hydrodynamic bores, which are front-type traveling waves that connect distinct asymptotic states, bifurcating from the trivial flat interface. 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 regime where gravity is order one. 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 introduce this phenomenon and explain how the analysis connects to a classical conjecture of von Kármán describing the structure of gravity currents near a rigid boundary.

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