Analysis Seminar

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Validity of the Nonlinear Schrodinger Approximation for the Two-Dimensional Water Wave Problem With and Without Surface Tension

Speaker: Wolf-Patrick DUELL, Universitaet Stuttgart

Location: Warren Weaver Hall 1302

Date: Thursday, February 27, 2020, 11 a.m.

Synopsis:

We consider the two-dimensional water wave problem in an infinitely long canal of
finite depth both with and without surface tension. In order to describe the evolution
of the envelopes of small oscillating wave packet-like solutions to this problem the
Nonlinear Schr ̈odinger equation can be derived as a formal approximation equation.
The rigorous justification of the Nonlinear Schr ̈odinger approximation for the water
wave problem was an open problem for a long time. In recent years, the validity
of this approximation has been proven by several authors only for the case without
surface tension.
In this talk, we present the first rigorous justification of the Nonlinear Schr ̈odinger
approximation for the two-dimensional water wave problem which is valid for the
cases with and without surface tension by proving error estimates over a physically
relevant timespan in the arc length formulation of the water wave problem. Our
error estimates are uniform with respect to the strength of the surface tension, as the
height of the wave packet and the surface tension go to zero.