Transverse Linear Stability of One-Dimensional Solitary Gravity Water Waves

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.