Finite Depth Gravity Water Waves in Holomorphic Coordinates

Abstract

Abstract. In this article we consider irrotational gravity water waves with finite bottom. Our goal is two-fold. First, we represent the equations in holomorphic coordinates and discuss the local well-posedness of the problem in this context. Second, we consider the small data problem and establish cubic lifespan bounds for the solutions. Our results are uniform in the infinite depth limit, and match our earlier infinite depth result in [8].

Recent advances on the global regularity for irrotational water waves

“…Sometimes it is possible to prove results going beyond the local theory, but not reach full global regularity. For example, starting with data of size ε in a standard Sobolev space, one can sometimes get ≈ ε −2 time of existence by proving a quartic energy inequality like (3.25) in cases when there are no significant quadratic resonances (see [75,82]). See also the recent work of Berti-Delort [22], where a combination of paradifferential analysis and ideas from KAM and normal forms theory was used to prove a significant long-time (≈ ε −N ) existence result for periodic 2D gravity-capillary waves (1D interface), for almost all choices of (g, σ).…”

Long Time Existence Results for Solutions of Water Waves Equations

“…x in (9) varies in R). For small, smooth and decaying initial data, global existence has been proved independently by Ifrim and Tataru [2017] and by Ionescu and Pusateri [2015a]. As far as we know, no (almost) global existence result is known for solutions of the full capillarygravity problem ((9) with g > 0; Ä > 0) in infinite depth, when the space dimension is equal to one.…”

On the Mathematical Description of Time-Dependent Surface Water Waves

“…Moreover, it is convenient to extend the functions η and φ holomorphically. This formulation was first used by Nalimov [90] and Ovsjannikov [92] and further developed by Dyachenko, Kuznetsov, Spector and Zakharov [51], Wu [125] as well as recently by Harrop-Griffiths, Hunter, Ifrim and Tataru [64,63]. An essential advantage of holomorphic coordinates is that in these coordinates the Dirichlet to Neumann operator is given in terms of the Hilbert transform in case of infinite depth of water and in terms of the operator K 0 with the symbol…”

“…They presented the details of the proof for the case of infinite depth, but, as they mentioned in [10], the proof works analogously in the case of finite depth. In holomorphic coordinates, local well-posedness of the 2-D water wave problem in Sobolev spaces was established by Harrop-Griffiths, Hunter, Ifrim and Tataru in the cases of infinite depth and no surface tension [64], infinite depth and surface tension but no gravity [66] as well as finite depth and no surface tension [63].…”