Long‐Term Regularity of 3D Gravity Water Waves

Zheng, Fan

doi:10.1002/cpa.21985

Cited by 7 publications

(4 citation statements)

References 90 publications

(172 reference statements)

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“…When the fluid velocity is irrotational (the vorticity curl u 0 = 0, a condition that is preserved by the evolution), the problem is called the (incompressible and irrotational) water wave problem which has attracted great attention for the long time existence. Previous works mostly focused on the case of an unbounded domain diffeomorphic to lower half-space or R d−1 × (−b, 0) and we refer to Wu [68,69] for the first breakthrough and numerous related works [19,20,4,30,15,23,22,24,25,64,73] 3 . See also [8] for the bounded domain case and [26,58] for some special cases when the vorticity is nonzero.…”

Section: An Overview Of Previous Resultsmentioning

confidence: 99%

Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

Luo¹,

Zhang²

2022

Preprint

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We consider 3D compressible isentropic Euler equations describing the motion of a liquid in an unbounded initial domain with a moving boundary and a fixed flat bottom at finite depth. The liquid is under the influence of gravity and surface tension and it is not assumed to be irrotational. We prove the local well-posedness by introducing carefully-designed approximate equations which are asymptotically consistent with the a priori energy estimates. The energy estimates yield no regularity loss and are uniform in Mach number. Also, they are uniform in surface tension coefficient if the Rayleigh-Taylor sign condition holds initially. We can thus simultaneously obtain incompressible and vanishing-surface-tension limits. The method developed in this paper is a unified and robust hyperbolic approach to free-boundary problems in compressible Euler equations. It can be applied to some important complex fluid models as it relies on neither parabolic regularization nor irrotational assumption. This paper joined with our previous works [45,46] rigorously proves the local well-posedness and the incompressible limit for a compressible gravity water wave with or without surface tension.

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Section: An Overview Of Previous Resultsmentioning

confidence: 99%

Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

Luo¹,

Zhang²

2022

Preprint

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“…While the whole structure of the proof resembles that in [24], here we aim not only to improve on known results on lifespans of two dimensional water waves, but also to show that the framework established in [24] is easily adaptable, and specifically, that the wealth of estimates already present in the literature, for example [2,5], can be readily assembled to yield a short proof of previously inaccessible results.…”

Section: Plan Of Actionmentioning

confidence: 99%

“…Let u = h + i|∇| 1/2 ψ, whose evolution equation is u t + iΛu = N, where the dispersion relation Λ = |∇| 1/2 and (see (6.1) and (4.43) in [24])…”

Section: Strichartz Estimatesmentioning

confidence: 99%

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Long-term regularity of 2D gravity water waves

Zheng

2022

Preprint

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The two dimensional gravity water wave problem concerns the motion of an incompressible fluid occupying half the 2D space and flowing under its own gravity. In this paper we study long-term regularity of solutions evolving from small but non-localized initial data.Our main result is that if the H s norm of the initial data is ǫ, where s ≫ 1 and ǫ ≪ 1, then the equation is wellposed at least for a time proportional to ǫ −4 , improving on the ǫ −3 lifespan obtained in [3,19]. We also study period water waves and show a lifespan bridging the gap between non-periodic waves and waves with a period of 1.

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On the wave turbulence theory of 2D gravity waves, I: Deterministic energy estimates

Deng,

Ionescu,

Pusateri

2024

Comm Pure Appl Math

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Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKEs) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as Schrödinger equations or multidimensional KdV‐type equations. However, our situation here is different since the water waves equations are quasilinear and solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue. We investigate solutions of the gravity water waves system in two dimensions. In the irrotational case, this system can be reduced to an evolution equation on the one‐dimensional interface, which is a large torus of size . Our first main result is a deterministic energy inequality, which provides control of (possibly large) Sobolev norms of solutions for long times, under the condition that a certain ‐type norm is small. This energy inequality is of “quintic” type: if the norm is , then the increment of the high‐order energies is controlled for times of the order , consistent with the approximate quartic integrability of the system. In the second paper in this sequence, we will show how to use this energy estimate and a propagation of randomness argument to prove a probabilistic regularity result up to times of the order , in a suitable scaling regime relating and . For our second main result, we combine the quintic energy inequality with a bootstrap argument using a suitable ‐norm of Strichartz‐type to prove that deterministic solutions with Sobolev data of size are regular for times of the order . In particular, on the real line, solutions exist for times of order . This improves substantially on all the earlier extended lifespan results for 2D gravity water waves with small Sobolev data.

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Long‐Term Regularity of 3D Gravity Water Waves

Cited by 7 publications

References 90 publications

Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

Compressible Gravity-Capillary Water Waves with Vorticity: Local Well-Posedness, Incompressible and Zero-Surface-Tension Limits

Long-term regularity of 2D gravity water waves

On the wave turbulence theory of 2D gravity waves, I: Deterministic energy estimates

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