## The Long Wave Approximation to the 3-D Capillary-Gravity Waves

### Jeudi 25 novembre 2010 14:15-15:15 - Zhang Ping - Academy of Mathematics, Chine

Résumé : In the regime of weakly transverse long waves, given long-waverninitial data, we prove that the nondimensionalized water wave system in an infinite strip under influence of gravity and surface tension on the upper free interface has a unique solution on $[0,T/\eps]$ for some $\eps$ independent of constant $T.$rnMoreover, we show that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev-Petviashvili (KP) equations for the strong and weak surface tension case, when the Bond number is positive and away from $\f13.$ In the case when the Bond number $\al=\frac13,$ the coefficients of the third order dispersion terms in $(KP)^\pm$rnequations vanish and the resulting equations become illposed. To capture the dispersive nature of the water-wave problem for this parameter regime, we modify the scaling in \eqrefscaling1 and thernapproximate equations. In fact, in this casern we shall prove that a new nondimensionalized water wave system has a unique solution on $[0,T/\eps^2]$ for some $\eps$ independent of constant $T.$ Moreover, we shall prove that on the same time interval, these solutions are accurately approximated by sums of solutions of two decoupled fifth-order Kadomtsev-Petviashvilirn$(KP^5th)^\pm$ equations.

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