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Transverse linear stability of one-dimensional solitary gravity water waves

Frédéric Rousset, Changzhen Sun

Abstract

In this paper, we establish the transverse linear asymptotic stability of one-dimensional small-amplitude solitary waves of the gravity water-waves system. More precisely, we show that the semigroup of the linearized operator about the solitary wave decays exponentially within a spectral subspace supplementary to the space generated by the spectral projection on continuous resonant modes. The key element of the proof is to establish suitable uniform resolvent estimates. To achieve this, we use different arguments depending on the size of the transverse frequencies. For high transverse frequencies, we use reductions based on pseudodifferential calculus, for intermediate ones, we use an energy-based approach relying on the design of various appropriate energy functionals for different regimes of longitudinal frequencies and for low frequencies, we use the KP-II approximation. As a corollary of our main result, we also get the spectral stability in the unweighted energy space.

Transverse linear stability of one-dimensional solitary gravity water waves

Abstract

In this paper, we establish the transverse linear asymptotic stability of one-dimensional small-amplitude solitary waves of the gravity water-waves system. More precisely, we show that the semigroup of the linearized operator about the solitary wave decays exponentially within a spectral subspace supplementary to the space generated by the spectral projection on continuous resonant modes. The key element of the proof is to establish suitable uniform resolvent estimates. To achieve this, we use different arguments depending on the size of the transverse frequencies. For high transverse frequencies, we use reductions based on pseudodifferential calculus, for intermediate ones, we use an energy-based approach relying on the design of various appropriate energy functionals for different regimes of longitudinal frequencies and for low frequencies, we use the KP-II approximation. As a corollary of our main result, we also get the spectral stability in the unweighted energy space.
Paper Structure (26 sections, 29 theorems, 453 equations)

This paper contains 26 sections, 29 theorems, 453 equations.

Table of Contents

  1. Introduction
  2. The line solitary waves
  3. Linearization of \ref{['sur-mov-uni']} about line solitary waves
  4. Outline of the proof and main difficulties
  5. Resonant modes
  6. Uniform resolvent estimate
  7. Spectral study in the regime of low transverse frequency
  8. KP-II approximation
  9. Existence of resonant modes--proof of Theorem \ref{['thm-resolmodes']}
  10. Spectral stability in the uniform high and intermediate transverse frequency regions.
  11. Proof of Proposition \ref{['prop-UTH']}
  12. Proof of Proposition \ref{['prop-ITF']}
  13. Proof of \ref{['es-uh']}
  14. Proof of \ref{['es-us']}
  15. Proof of \ref{['es-ur']}
  16. ...and 11 more sections

Key Result

Theorem 1.1

Assuming $\gamma=1-\epsilon^2,$ there exists $\epsilon_0$ such that for every $\epsilon\in (0, \epsilon_0)$ there is a stationary solution $\zeta_{c}(x), \varphi_{c}(x)$ (independent of $y$), given by where $\Theta$ and $\Phi$ satisfy the following: (1) There exists $d>0$, for any integers $k\geq 0 , \, \ell\geq 1,$ there exists $C_k, \, C_{\ell}$ such that: (2) It holds that when $\epsilon=0,$

Theorems & Definitions (50)

  • Theorem 1.1: Restatement of Beale's result Beale
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • ...and 40 more