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Geometric approaches to water waves and free surface flows - 4

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Free Boundary Problems and Related Topics

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:

[1] Buffoni, B.; Toland, J. F. Analytic theory of global bifurcation. An introduction. Princeton Series in Applied Mathematics. Princeton University Press, Princeton, N.J., 2003.

[2] Constantin, A. Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS -NSF Regional Conference Series in Applied Mathematics, 81. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, P.A., 2011.

[3] Constantin, A.; Strauss, W. Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math. 57 (2004), no. 4, 481—527.

[4] Varvaruca, E. On the existence of extreme waves and the Stokes conjecture with vorticity. J. Differential Equations 246 (2009), no. 10, 4043—4076.

[5] Varvaruca, E.; Weiss, G. S. A geometric approach to generalized Stokes conjectures. Acta Math. 206 (2011), no. 2, 363—403.

[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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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