Abstract

These lectures aim to present a new geometric approach to the asymptotic behaviour near singularities in some classical free-boundary problems in fluid dynamics. We start by introducing the problems and providing an outline of the methods that have been used to prove existence of solutions. We then present a modern proof, using monotonicity formulas and frequency formulas, of the famous Stokes conjecture from 1880, which asserts that at any stagnation point on the free surface of a two-dimensional 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 restrictive assumptions and by somewhat ad-hoc methods.) We then explain how the methods extend to the case of two-dimensional steady gravity water waves with vorticity. Finally, we show how the same methods can be adapted to describe the asymptotic behaviour near singularities in the problem of steady three-dimensional axisymmetric free surface flows with gravity.

References:

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[6] Varvaruca, E.; Weiss, G. S. The Stokes conjecture for waves with vorticity. Ann. Inst. H. Poincar Anal. Non Linaire 29 (2012), no. 6, 861—885.

[7] Varvaruca, E.; Weiss, G. S. Singularities of steady axisymmetric free surface flows with gravity, to appear in Comm. Pure Appl. Math., http://arxiv.org/abs/1210.3682.