Conditional energetic stability of gravity solitary waves in the presence of weak surface tension

Abstract

For a sequence of values of the total horizontal impulse that converges to $0$, there are solitary waves that minimise the energy in a given neighbourhood of the origin in $W^{2,2}({\mathbb R})$. The problem arises in the framework of the classical Euler equation when a two-dimensional layer of water above an infinite horizontal bottom is considered, at the surface of which solitary waves propagate under the action of gravity and weak surface tension. The adjective "weak" refers to the Bond number, which is assumed to be sub-critical ($< 1/3$).

This extends previous results on the conditional energetic stability of solitary waves in the super-critical case, namely those by A. Mielke ([On the energetic stability of solitary water waves, Philos. Trans. Roy. Soc. London Ser. A 360 (2002), 2337–2358]) and by the author ([Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation, Arch. Rational Mech. Anal.]). Like in the latter, the method is based on direct minimisation and concentrated compactness, but without relying on "strict sub-additivity", which is still unsettled in the present case. Instead, a complete and careful analysis of minimising sequences is performed that allows us to reach a conclusion, based only on the non-existence of "vanishing" minimising sequences. However, in contrast with [Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation, Arch. Rational Mech. Anal.], we are unable to prove the existence of minimisers for all small values of the total horizontal impulse.

In fact more is needed to get stability, namely that every minimising sequence has a subsequence that converges to a global minimiser, after possible shifts in the horizontal direction. This will be obtained as a consequence of the analysis of minimising sequences. Then exactly the same argument as in [Existence and conditional energetic stability of capillary-gravity solitary water waves by minimisation, Arch. Rational Mech. Anal.] gives conditional energetic stability and is therefore not repeated.