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SUMMARY:Geometric approaches to water waves and free surface flows - 4 - V
 arvaruca\, E (University of Reading)
DTSTART:20140110T100000Z
DTEND:20140110T110000Z
UID:TALK49704@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:These lectures aim to present a new geometric approach to the 
 asymptotic behaviour near singularities in some classical free-boundary pr
 oblems in fluid dynamics. We start by introducing the problems and providi
 ng an outline of the methods that have been used to prove existence of sol
 utions. We then present a modern proof\, using monotonicity formulas and f
 requency formulas\, of the famous Stokes conjecture from 1880\, which asse
 rts 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 somewha
 t ad-hoc methods.) We then explain how the methods extend to the case of t
 wo-dimensional steady gravity water waves with vorticity. Finally\, we sho
 w how the same methods can be adapted to describe the asymptotic behaviour
  near singularities in the problem of steady three-dimensional axisymmetri
 c free surface flows with gravity.\n\nReferences: \n\n[1] Buffoni\, B.\; T
 oland\, J. F. Analytic theory of global bifurcation. An introduction. Prin
 ceton Series in Applied Mathematics. Princeton University Press\, Princeto
 n\, N.J.\, 2003.\n\n[2] Constantin\, A. Nonlinear water waves with applica
 tions to wave-current interactions and tsunamis. CBMS-NSF Regional Confere
 nce Series in Applied Mathematics\, 81. Society for Industrial and Applied
  Mathematics (SIAM)\, Philadelphia\, P.A.\, 2011.\n\n[3] Constantin\, A.\;
  Strauss\, W. Exact steady periodic water waves with vorticity. Comm. Pure
  Appl. Math. 57 (2004)\, no. 4\, 481--527.\n\n[4] Varvaruca\, E. On the ex
 istence of extreme waves and the Stokes conjecture with vorticity. \nJ. Di
 fferential Equations 246 (2009)\, no. 10\, 4043--4076.\n\n[5] Varvaruca\, 
 E.\; Weiss\, G. S. A geometric approach to generalized Stokes conjectures.
  Acta Math. 206 (2011)\, no. 2\, 363--403.\n\n[6] Varvaruca\, E.\; Weiss\,
  G. S. The Stokes conjecture for waves with vorticity. Ann. Inst. \nH. Poi
 ncar Anal. Non Linaire 29 (2012)\, no. 6\, 861--885.\n\n[7] Varvaruca\, E.
 \; Weiss\, G. S. Singularities of steady axisymmetric free surface flows w
 ith gravity\, to appear in Comm. Pure Appl. Math.\, http://arxiv.org/abs/1
 210.3682.\n
LOCATION:Seminar Room 1\, Newton Institute
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