Almost global wellposedness of the 2-D full water wave problem
This interesting paper is devoted to the problem of global in time existence and uniqueness of solutions of the \(2-D\) infinite depth full water wave equation. When surface tension is zero, the motion of the fluid is described by \[ \mathbf v_t+v\cdot \nabla v=-\mathbf k -\nabla P \quad \text{on } \Omega (t), \;t\geq 0,\tag{1} \] \[ \text{div } \mathbf v =0, \quad \text{curl }\mathbf v=0 \quad \text{on } \Omega (t), \;t\geq 0,\tag{2} \] \[ P=0 \quad \text{on } \Sigma (t),\tag{3} \] \[ (1, \mathbf v) \text{ is tangent to the free surface } (t,\Sigma (t)),\tag{4} \] where \(\mathbf v\) is the fluid velocity, \(P\) is the fluid pressure, \(\Sigma (t)\) (\(t\) - the time) is the free interface, \(\Omega (t)\) the region occupied by the fluid, and \(\mathbf k\) is the gravitational field (the unit vector pointing in the upward vertical direction). The author investigates the global in time behavior of solutions of the problem (1)--(4). It is known that there exists a solution defined in a time interval \([0,T/\epsilon ]\) for initial data of the form \(\epsilon\Psi \), where \(T\) depends only on \(\Psi \). An interesting result shown here is that for the same initial data there exists a unique solution in the interval \([0,T/\epsilon ]\). It turns out that there exists quantities \(\Theta \) that are the traces on the interface \(\Sigma (t)\) of some holomorphic or almost holomorphic functions in the air region \(\Omega (t)^c\). These quantities expressed in a new coordinate system satisfy equations of the form \(\partial_t^2\Theta -i\partial_{\alpha }\Theta =G\), where \(G\) consists of nonlinear terms of only cubic and higher orders. Using these equations one can prove existence and uniqueness of the solution to the problem (1)--(4) for the above stated initial data.
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