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CATEGORIES:Fluid Mechanics (DAMTP)
SUMMARY:Wave breaking and ill-posedness of the perturbatio
n theory for water waves - Pavel Lushnikov (Unive
rsity of New Mexico)
DTSTART;TZID=Europe/London:20110114T160000
DTEND;TZID=Europe/London:20110114T170000
UID:TALK28848AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/28848
DESCRIPTION:Breaking of water waves is a dramatic phenomenon\,
which often\noccurs in nature. Wave breaking can
characterized by finite time\nsingularity formatio
n in solutions of dynamic equations. In the talk w
e give three examples of wave breaking. The first
is related to foam formation on crests of sea wav
es due to Kelvin-Helmholtz instability of the inte
rface between two ideal fluids. Evolution of inter
face surface is described by a nonlinear (2+1)-dim
ensional Klein-Gordon equation. A proof of singul
arity formation in a finite time is given. Our res
ults agree with the sharp dependence on wind veloc
ity of the fraction of sea surface area covered by
foam as obtained from satellite and airplane obse
rvations. The second example concerns the integrab
le dynamics of the interface between a light visco
us fluid with Stokes flow and a heavy ideal fluid.
Surface evolution is determined from the motion o
f complex singularities (poles) of two complex Bu
rgers equations. The interface loses its smoothnes
s if poles reach the interface. In the third examp
le we show that sometimes wave breaking does not r
eally occur. We consider the Hamiltonian form of t
he water wave equations for the free surface motio
n and show that they are ill-posed and formally wa
ve breaking should happen in arbitrary small time.
However we found that these equations become well
-posed after a canonical transformation to new var
iables and no wave breaking actually occurs. Impli
cations of the new variables for numerical simulat
ions of ocean dynamics are discussed.
LOCATION:Note! MR3 Centre for Mathematical Sciences\, Wilbe
rforce Road\, Cambridge
CONTACT:Doris Allen
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