CAMS Colloquium: The quartic integrability and long time existence of steep water waves in 2d

Monday, September 20 at 3:30pm to 4:30pm

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Virtual Event

Sijue Wu, University of Michigan

Abstract: It is known since the work of Dyachenko & Zakharov in 1994 that for weakly nonlinear 2d infinite 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 integrability from a different angle.  We construct a sequence of energy functionals $E_j(t)$,  directly in the physical space,  which are explicit in the Riemann mapping variable and involve material derivatives of order $j$ of 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 norm, 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 the 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 this 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 the slope bof the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses  and initial velocities to have arbitrary large magnitudes.


Dial-In Information

Zoom Meeting:
Meeting ID: 947 2844 8929
Passcode: 196455

and CAMS Colloquium chat room:
Meeting ID: 937 2442 0445
Passcode: 019455

Event Type

Lecture / Talk / Workshop

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