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We derive here a new stability criterion for two-fluid interfaces. Thiscriterion ensures the existence of "stable" local solutions that do no breakdown too fast due to Kelvin-Helmholtz instabilities. It can be seen both as atwo-fluid generalization of the Rayleigh-Taylor criterion and as a nonlinearversion of the Kelvin stability condition. We show that gravity can control theinertial effects of the shear up to frequencies that are high enough for thesurface tension to play a relevant role. This explains why surface tension is anecessary condition for well-posedness while the (low frequency) main dynamicsof interfacial waves is unaffected by it. In order to derive a practicalversion of this criterion, we work with a nondimensionalized version of theequations and allow for the possibility of various asymptotic regimes, such asthe shallow water limit. This limit being singular, we have to derive a newsymbolic analysis of the Dirichlet-Neumann operator that includes an infinitelysmoothing "tail" accounting for the contribution of the bottom. We thenvalidate our criterion by comparison with experimental data in two importantsettings: air-water interfaces and internal waves. The good agreement weobserve allows us to discuss the scenario of wave breaking and the behavior ofwater-brine interfaces. We also show how to rigorously justify two-fluidasymptotic models used for applications and how to relate some of theirproperties to Kelvin-Helmholtz instabilities.

A stability criterion for two-fluid interfaces and applications