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Existence and Conditional Energetic Stability of Capillary-Gravity Solitary Water Waves by Minimisation

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Abstract.

Firstly, the two-dimensional stationary water-wave problem is considered. Existence of capillary-gravity solitary waves is proved by minimising a functional related to Smale’s amended potential. We first establish the existence of periodic solutions of arbitrarily large periods, leading to a minimising sequence in L2(ℝ) that stays away from the boundary of the neighbourhood of 0 ∈ W2,2(ℝ) in which the analysis is carried out. With the help of the concentration-compactness principle, we then show that every minimising sequence has a subsequence that, after possible shifts in the propagation direction, converges in L2(ℝ) to a minimiser. Secondly, for the evolutionary problem, we prove that the set of minimal solitary waves as a whole is energetically conditionally stable. “Energetically” means that the distance to the set of all minimisers is defined in terms of the total energy, and “conditionally” means that we consider solutions to the evolutionary problem that do not explode instantaneously but could perhaps explode in finite time (e.g., via the explosion of another norm). We work in some bounded set in W2,2(ℝ) that contains the quiescent state and we are not interested in the fate of solutions that leave this set.

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  1. Section de mathématiques (IACS) École, Polytechnique Fédérale, Lausanne, CH-1015, Boris

    B. Buffoni

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  1. B. Buffoni
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Communicated by A. Mielke

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Buffoni, B. Existence and Conditional Energetic Stability of Capillary-Gravity Solitary Water Waves by Minimisation. Arch. Rational Mech. Anal. 173, 25–68 (2004). https://doi.org/10.1007/s00205-004-0310-0

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