Singularities of steady free surface water flows
Seminar Room 1, Newton Institute
We present some recent results, based on a geometric analysis approach, that provide a characterization of all possible singularities in two related free-boundary problems in hydrodynamics: that of steady two-dimensional gravity water waves and that of steady three-dimensional axisymmetric water flows under gravity. In the 2D problem, we outline a modern proof, using a blow-up analysis based on a monotonicity formula and a frequency formula, of the Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a steady irrotational gravity water wave, the wave profile necessarily has lateral tangents enclosing a symmetric angle of 120 degrees. (This result was first proved in the 1980s under some restrictive assumptions and by somewhat
ad-hoc methods.) The new approach extends easily to the case when the effects of vorticity in the flow are included. Moreover, we explain how the methods can be adapted to the 3D axisymmetric problem, where several different types of singularities are possible, depending on whether one is dealing with a stagnation point, a point on the axis of symmetry, or both (in the case of the origin). For example, in the case of the origin, there are only two possible types of singular asymptotic behaviour: one is a conical singularity called ``Garabedian corner flow", and the other is a flat degenerate point; while in the case of points on the axis of symmetry different from the origin, cusps are the only possible singularities. These results were obtained in joint works with Georg Weiss (Dusseldorf).
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