Bifurcation analysis for surface waves generated by wind (Q2783716)

In two papers published in [SIAM J. Math. Anal. 28, No. 5, 1135--1157 (1997; Zbl 0889.35075) and Commun. Partial Differ. Equations 25, No. 5--6, 887--901 (2000; Zbl 0955.35062)], the author investigated the generation of waves by the wind in an incompressible viscous fluid (Navier-Stokes equations). The wind effect results in the pressure difference between backward and forward forces on the wave. In the above quoted papers the author showed that at a critical speed of the wind, there appears a Hopf bifurcation and, also, traveling wave solutions. In the present paper the author is interested in the effect of the system's nonlinearity, given by the convective term and by the change of domain. The two-dimensional Navier-Stokes equations and a model for the wind action, which creates pressure distributions along the surface and exerts tangential forces, are the start points. The paper is divided in the following parts: 1. the introduction, where the boundary value problem governing the flow is established; 2. a linear problem is derived from it, and explicit formulae are derived for bifurcation parameters \(\gamma\) and \(c\) (the speed of the wind and wave velocity, respectively) and for the corresponding eigenfunctions; 3. the fluid domain with unknown free boundary is transformed into a reference fix domain with the corresponding boundary value problem; 4. the bifurcation branch is studied in the nonlinear case, a tangential velocity appears along the free boundary, and this shear-flow grows with the wave; 5. the discretization in finite elements of the flow, and the interpretation of the numerical results