Ill-posedness of truncated series models for water waves

f6-largeDavid Ambrose, on Thusday afternoon 31 July, gave a talk on ill-posedness of truncated series modes for water waves.  The models under study are obtained by expanding the Dirichlet-Neumann operator in the Zakharov-Craig-Sulem formulation of gravity water waves.  Two models were discussed in the talk: WW2 (a quadratic truncation) and WW3 (a cubic reduction).  The key to the theory was showing that u_t = L(u) where the symmetric operator L is a Hilbert transform composed with a space derivative is backward parabolic.  Then u_t = L(u)^2 – u_x^2 was studied and shown to be ill-posed. Numerical results then support the ill-posedness of the full system including the free surface.  Numerical results for WW3 were also presented, suggesting ill-posedness in this case as well.  These results were reported in Ambrose, Bona & Nicholls (2014). Adding viscosity, as in Ambrose, Bona & Nicholls (2012) recovered well-posedness. The addition of surface tension was also discussed: analysis suggests it should not be sufficient to regularise whereas numerics is suggesting that it does regularise. A video is available here.

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