Description

• Tytuł:
  Conditional energetic stability of gravity solitary waves in the presence of weak surface tension
• Twórca:
  Buffoni, Boris
• Opis:
  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.
• Dostępność obiektu:
  free access
• Prawa:
  Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
• Data:
  2005.03.01
• Typ:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Język:
  English
• Źródło:
  Topological Methods in Nonlinear Analysis; Vol 25, No 1 (March 2005); 41 - 68;
  1230-3429
• Wydawca:
  Nicolaus Copernicus University in Toruń
• Relacja: https://apcz.umk.pl/TMNA/article/view/TMNA.2005.003/9255
• Temat i słowa kluczowe:
  Capilary-gravity water waves;
  solitary waves;
  stability;
  variational methods;
  76B15;
  76B25;
  76E17;
  35J20
• Identyfikator: https://apcz.umk.pl/TMNA/article/view/TMNA.2005.003
• Dostawca danych:
  Akademicka Platforma Czasopism
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