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SUMMARY:Wave breaking and ill-posedness of the perturbation theory for wat
 er waves -  Pavel Lushnikov (University of New Mexico)
DTSTART:20110114T160000Z
DTEND:20110114T170000Z
UID:TALK28848@talks.cam.ac.uk
CONTACT:Doris Allen
DESCRIPTION:Breaking of water waves is a dramatic phenomenon\, which often
 \noccurs in nature. Wave breaking can characterized by finite time\nsingul
 arity formation in solutions of dynamic equations. In the talk we give thr
 ee examples of wave breaking.  The first is related to foam formation on c
 rests of sea waves due to Kelvin-Helmholtz instability of the interface be
 tween two ideal fluids. Evolution of interface surface is described by a n
 onlinear (2+1)-dimensional Klein-Gordon equation.  A proof of singularity 
 formation in a finite time is given. Our results agree with the sharp depe
 ndence on wind velocity of the fraction of sea surface area covered by foa
 m as obtained from satellite and airplane observations. The second example
  concerns the integrable dynamics of the interface between a light viscous
  fluid with Stokes flow and a heavy ideal fluid. Surface evolution is dete
 rmined from the motion of complex singularities (poles) of  two complex Bu
 rgers equations. The interface loses its smoothness if poles reach the int
 erface. In the third example we show that sometimes wave breaking does not
  really occur. We consider the Hamiltonian form of the water wave equation
 s for the free surface motion and show that they are ill-posed and formall
 y wave breaking should happen in arbitrary small time. However we found th
 at these equations become well-posed after a canonical transformation to n
 ew variables and no wave breaking actually occurs. Implications of the new
  variables for numerical simulations of ocean dynamics are discussed.
LOCATION:Note! MR3 Centre for Mathematical Sciences\, Wilberforce Road\, C
 ambridge
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