In this talk I will present two results in fluid mechanics whose common ingredient is bifurcation theory. The first one shows the existence of stationary solutions to the surface quasi-geostrophic equation, a 2D model for the 3D incompressible Euler equations used in geophysics. The second one is an example of application of a new method, using simple ideas and computer-assisted estimates, that allows the construction of endpoints of global bifurcation branches.

Specifically, we construct highest (limiting) waves to the Whitham equation and show that the profile is convex between consecutive stagnation points. This answers a conjecture of Ehrnström and Wahlén.

Joint work with Alberto Enciso and Bruno Vergara.