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On a local existence theorem in the theory of surface waves of finite amplitude. (English. Russian original) Zbl 0641.76008
Sov. Math., Dokl. 35, No. 3, 647-650 (1987); translation from Dokl. Akad. Nauk SSSR 294, No. 6, 1289-1292 (1987).
We consider the problem of steady-state periodic waves of finite amplitude on the surface of a heavy fluid of infinite depth. We de- dimensionalize the problem, referring lengths to \(\lambda\) as unit, where \(\lambda\) is the wave length, and velocities to \(\sqrt{g\lambda}\). In a coordinate system in which the waves are at rest, the motion is stationary and potential. Let \(\phi\) be a velocity potential, and let \(\psi\) be the flow function; we choose the constant in the definition of the function \(\zeta =\phi +i\psi\) so that the equality \(\zeta =0\) holds at one of the wave crests.
Let \(L_ 2[S^ 1]\) be the Hilbert space of \(2\pi\)-periodic functions, and let \({\mathcal L}_ 2\) be the subspace of \(L_ 2[S^ 1]\) consisting of even functions. In \(L_ 2[S^ 1]\) we define the Hilbert operator H, the differentiation operator D, and the operator \(J=HD\). We denote by \({\mathcal W}^ r_ 2\) the space of \(2\pi\)-periodic even functions with norm \(| f|^ 2_ r=\| f\|^ 2+\| D^ rf\|^ 2\), where \(\| \cdot \|\) is the norm in \(L_ 2[S^ 1]\). In a previous note [see the review above (Zbl 0641.76007)] we proved that the problem of periodic progressive waves on the surface of a heavy fluid of infinite depth is equivalent to the problem of solving the equation (*) \(J(y^ 2)+(y-c^ 2)J(y)+y=0\) in some space \({\mathcal W}^ r_ 2.\)
An existence theorem for nonlinear progressive waves of small amplitude is proved in the familiar papers of Nekrasov and Levi-Civita. Essentially Nekrasov and Levi-Civita considered the problem of bifurcation of the zero solution of (*), and they constructed a solution of small amplitude branching off from \(y\equiv 0\). It would seem that there is no reason to return to this problem, but unfortunately the old papers give no estimate of the domain of applicability of the theorems. At the same time, if the computations are configured to address this question they cannot be carried out for very small values of the amplitude, since the differential of the mapping F:\({\mathcal W}_ 1^{r+1}\to {\mathcal W}^ r_ 2\), where \(F=F(y,c)\) is the left side of (*), will be an operator close to a degenerate operator when \(\| y\|\) is small and the parameter \(c^ 2\) is closed to its bifurcation value. We are therefore forced to turn again to a local existence theorem.

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M99 Basic methods in fluid mechanics
35Q30 Navier-Stokes equations