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 {\it 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
([< i> On the energetic stability of solitary water waves< /i> , Philos. Trans. Roy. Soc.
London Ser. A < b> 360< /b> (2002), 2337–2358]) and
by the author ([< i> Existence and conditional energetic stability of capillary-gravity
solitary water waves by minimisation< /i> , 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 [< i> Existence and conditional energetic stability of capillary-gravity
solitary water waves by minimisation< /i> , Arch. Rational Mech. Anal.],
we are unable to prove the existence
of minimisers for < i> all< /i> 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 [< i> Existence and conditional energetic stability of capillary-gravity
solitary water waves by minimisation< /i> , Arch. Rational Mech. Anal.] gives conditional energetic
stability and is therefore
not repeated.