Well-posedness of the water-waves equations

By David Lannes. 2005

478 citations recorded by the source corpus.

1.1. Presentation of the problem. The water-waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect, incompressible, irrotational fluid under the influence of gravity. In this paper, we restrict our attention to the case when the surface is a graph parameterized by a function ?(?,X), where t denotes the time variable and X = (Xi,... ,Xd) Rd the horizontal spatial variables. The method developed here works equally well for any integer d > 1, but the only physically relevant cases are of course d = 1 and d ? 2. The layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time-independent function b(X). We denote by Q,t the fluid domain at time t. The incompressibility of the fluids is expressed by (1.1) div V = 0 in Qu t > 0,

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/02e6ee1ea1c5a809c42a655f8796d63ddeffe926

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Citations in the corpus

Asymptotic Models for Internal Waves (2007, methodology): “The key ingredient in the proof is an explicit formula of the derivative of the mapping ζ → V μ[εζ ]ψ , which generalizes the formula obtained in [26] for the shape derivative of Dirichlet–Neumann operators.”
Almost Exponential Decay of Periodic Viscous Surface Waves without Surface Tension (2013, unknown): “Ē−2N (0) + ∫ t 0 Eθ2ND2N + εD2N + ε−8N−1D̄02N .”
Two dimensional water waves in holomorphic coordinates II: global solutions (2014, unknown): “Without being exhaustive, we list some of the more recent references [1, 2, 4, 5, 7, 8, 15, 17, 20, 25].”
Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum (2012, background): “We refer the reader to [1,9,19,21,32,34,41,42,45], and [46] for a partial list of papers on this topic.”
Nonlinear fractional Schrödinger equations in one dimension (2012, unknown): “See for example [21, 19, 25].”
Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations (2012, methodology): “The easiest way to fix this difficulty is to to take a smoothing diffeomorphism as in [48], [33].”
Long Wave Approximations for Water Waves (2005, background): “Some results concerning the Cauchy problem wherein the free surface is specified for all values of X and the velocity potential is specified everywhere in the resulting flow domain, both at a given instant of time, are available (see, for example, [11], [17], [19], [21], [22], [ 16 ], and the refere”
Time quasi-periodic gravity water waves in finite depth (2017, methodology): “The matrix Z in (7.12) amounts to introduce, as in Lannes [45]-[46], a linearized version of the good unknown of Alinhac, working with the variables (η, ς) with ς := ψ −Bη, instead of (η, ψ).”
Decay of viscous surface waves without surface tension (2010, unknown): “22”
Geometry and a priori estimates for free boundary problems of the Euler's equation (2006, unknown): “There are many other works on this problem we mention the work of D. Lannes [ LA05 ], T.”
Finite time singularities for the free boundary incompressible Euler equations (2011, background): “Lannes [21] considered the case in which water is moving over a fixed bottom.”
Well-posedness of the free-surface incompressible Euler equations with or without surface tension (2005, methodology): “If one uses a Nash-Moser approach without performing the analysis of [6], then one obtains results with much higher regularity requirements than necessary, as for instance in [11] for the irrotational water-wave problem without surface tension.”
On the Cauchy problem for gravity water waves (2012, methodology): “Secondly, it was proved by Lannes ([37]) that the Taylor’s assumption is satisfied under a smallness assumption on the curvature of the bottom (provided that the bottom is at least C2).”
Global solutions for the gravity water waves equation in dimension 3 (2009, result): “ other works on local well posedness: we mention the work of W. Craig [CR85], T. J. Beal, T. Hou, and J. Lowengrub [BHL93], M. Ogawa and A. Tani [OT02], G. Schneider and E. C. Wayne [SW02], D. Lannes [LA05], D. Ambrose and N. Mamoudi [AM06] [AM07], and P. Zhang and Z. Zhang [ZZ06]. We also mention t”
Paralinearization of the Muskat Equation and Application to the Cauchy Problem (2019, methodology): “ting the diﬀerence of two solutions. With regards to the existence, we construct solutions to the Muskat equation as limits of solutions to a sequence of approximate nonlinear systems, following here [1, 2, 32,33]. We split the analysis in three parts. (1) Firstly, we prove that the Cauchy problem f”
On the free boundary problem of three‐dimensional incompressible Euler equations (2008, background): “Lannes [11] considered the same problem in the case of finite depth.”
Solvability of the free boundary value problem of the Navier-Stokes equations (2010, background): “For the recent works on the Euler equations for the rotational case, see [2, 7, 11, 14, 22].”