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Nonlinear stationary surface waves above an underwater ridge. (Russian, English) Zbl 1017.35083
Sib. Mat. Zh. 43, No. 4, 783-810 (2002); translation in Sib. Math. J. 43, No. 4, 626-650 (2002).
The author studies traveling gravitation-capillary waves of small amplitude above an underwater ridge in an exact nonlinear statement for an irrotational flow of an ideal incompressible fluid. A sufficient condition for existence of a smooth solution is the inverse proportional dependence between the surface tension and wave frequency. The problem is studied in special Banach spaces (Hardy-type classes) of functions periodic in the variable along the underwater ridge and decreasing exponentially in the perpendicular direction. The author reduces this problem to a nonlinear problem of bifurcation theory. Its specific feature is that the linear homogeneous problem is irregular: the corresponding eigenfunction vanishes as the small parameter characterizing the deviation of the bottom from a horizontal plane tends to zero. A bifurcation comes from an approximate solution that is obtained by finite-rank approximation of an operator such that the linear problem is reduced to finding its fixed point [D. S. Kuznetsov, Sib. Math. J. 42, 668-684 (2001; Zbl 0986.47014)].
On assuming that the elevation of the underwater ridge is small, an existence theorem is proven for smooth solutions to equations of an ideal incompressible fluid which are periodic in the variable directed along the underwater ridge and decreasing exponentially in the perpendicular direction with small positive exponent in the perpendicular direction.
The author’s method consists in successive reduction of the three-dimensional problem to some one-dimensional problem and proving solvability of the resulting nonlinear pseudodifferential problem.
MSC:
35Q05 Euler-Poisson-Darboux equations
76B55 Internal waves for incompressible inviscid fluids
35B35 Stability in context of PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
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