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Unstable surface waves in running water. (English) Zbl 1170.35080

Commun. Math. Phys. 282, No. 3, 733-796 (2008); erratum ibid. 318, No. 3, 857-861 (2013).
This paper provides an extensive study of periodic free-surface water waves under gravity. Specifically, the authors investigate the stability of periodic waves propagating downstream at constant speed over a shear flow of finite depth. They begin by examining the stability of shear flows for a flat surface. In this case, they obtain a sharp criterion of linear instability for a class of shear flows with inflection points and determine the maximal unstable wave number. The authors conclude that the free surface has a destabilizing effect when compared with the rigid-wall setting that has been investigated by Z.-W Lin [SIAM J. Math. Anal. 35, 318–356 (2003; Zbl 1126.76323)] and others (see references therein). They then establish the bifurcation of (nontrivial) periodic travelling waves at all wave numbers for unstable shear flows. Moreover, they demonstrate the linear instability of small nontrivial waves that appear after bifurcation (at an unstable wave number of the underlying shear flow). To accomplish this, they invoke a novel formulation of the linearized water-wave problem and a perturbation argument. Finally, the authors show that vorticity has a subtle influence on the stability of free-surface water waves. They do so by constructing an example of the background shear flow of unstable small-amplitude periodic travelling waves for an arbitrary vorticity strength and an arbitrary depth. The article is complemented by an extensive bibliography on the mathematical theory of gravity water waves, covering both historical and recent results.

MSC:

35Q35 PDEs in connection with fluid mechanics
35R35 Free boundary problems for PDEs
76E05 Parallel shear flows in hydrodynamic stability
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35B32 Bifurcations in context of PDEs

Citations:

Zbl 1126.76323
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References:

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