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CATEGORIES:Lecture / Talk / Workshop
DESCRIPTION:Sijue Wu\, University of Michigan\n\nAbstract: It is known sinc
e the work of Dyachenko & Zakharov in 1994 that for weakly nonlinear 2d inf
inite depth water waves\, there are no 3-wave interactions and all of the
4-wave interaction coefficients vanish on the non-trivial resonant manifold
. In this talk I will present a recent result that proves this partial inte
grability from a different angle. We construct a sequence of energy functi
onals $E_j(t)$\, directly in the physical space\, which are explicit in t
he Riemann mapping variable and involve material derivatives of order $j$ o
f the solutions for the 2d water wave equation\, so that $\frac d{dt} E_
j(t)$ is quintic or higher order. We show that if some scaling invariant n
orm\, and a norm involving one spacial derivative above the scaling of the
initial data are of size no more than $\epsilon$\, then the lifespan of th
e solution for the 2d water wave equation is at least of order $O(\epsilon^
{-3})$\, and the solution remains as regular as the initial data during thi
s time. If only the scaling invariant norm of the data is of size $\epsilon
$\, then the lifespan of the solution is at least of order $O(\epsilon^{-5/
2})$. Our long time existence results do not impose size restrictions on th
e slope bof the initial interface and the magnitude of the initial velocity
\, they allow the interface to have arbitrary large steepnesses and initia
l velocities to have arbitrary large magnitudes.
DTEND:20210920T233000Z
DTSTAMP:20211201T153808Z
DTSTART:20210920T223000Z
LOCATION:
SEQUENCE:0
SUMMARY:CAMS Colloquium: The quartic integrability and long time existence
of steep water waves in 2d
UID:tag:localist.com\,2008:EventInstance_37837427533528
URL:https://calendar.usc.edu/event/cams_colloquium_the_quartic_integrabilit
y_and_long_time_existence_of_steep_water_waves_in_2d
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