Well-posedness of the free-surface incompressible Euler equations with or without surface tension

By Daniel Coutand, Steve Shkoller. 2005

413 citations recorded by the source corpus.

We provide a new method for treating free boundary problems in perfect fluids, and prove local-in-time well-posedness in Sobolev spaces for the free-surface incompressible 3D Euler equations with or without surface tension for arbitrary initial data, and without any irrotationality assumption on the fluid. This is a free boundary problem for the motion of an incompressible perfect liquid in vacuum, wherein the motion of the fluid interacts with the motion of the free-surface at highest-order

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cef5d50cd2c36cc8810c8357e42e1799a7fe301f

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Citations in the corpus

Almost Exponential Decay of Periodic Viscous Surface Waves without Surface Tension (2013, neutral): “Ē−2N (0) + ∫ t 0 Eθ2ND2N + εD2N + ε−8N−1D̄02N .”
Two Dimensional Water Waves in Holomorphic Coordinates (2014, neutral): “Without being exhaustive, we list some of the more recent references [1,2,4,5,7,14,17,19,22,29].”
On the Euler equations of incompressible fluids (2007, methodology): “The modern well-posedness theory for free surfaces uses harmonic analysis and geometric PDEs ([3], [4], [35], [63], [65], [66], [102], [126], [127]) and is a subject in full swing.”
Well-Posedness in Smooth Function Spaces for the Moving-Boundary Three-Dimensional Compressible Euler Equations in Physical Vacuum (2012, methodology): “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.”
Almost global wellposedness of the 2-D full water wave problem (2009, background): “There has been much work recently concerning water waves with additional effects such as a nonzero surface tension, a bottom, and a bounded nonzero vorticity in the fluid region [1, 6, 7 , 14, 24, 26, 28, 31, 41].”
Finite time singularities for the free boundary incompressible Euler equations (2011, neutral): “See Christodoulou-Lindblad [12], Lindblad [23], Coutand-Shkoller [16], Shatah-Zeng [28] and Zhang-Zhang [35].”
On the Cauchy problem for gravity water waves (2012, background): “) by Shatah-Zeng [47, 48, 49], Christodoulou–Lindblad [20] or Lindblad [39] seem to at least require bounded curvature assumptions (see also [23] where a logarithmic divergence is allowed).”
Global small solutions to 2-D incompressible MHD system (2013, neutral): “1) As in [11], [12], [13], [17], [27], [30], [32] and various earlier references therein, we shall use the Lagrangian formulation.”
On the free boundary problem of three‐dimensional incompressible Euler equations (2008, methodology): “We should also mention the most recent interesting results on this topic by D. Coutand and S. Shkoller [5] and J. Shatah and C. Zeng [18].”
Global solutions for the gravity water waves equation in dimension 3 (2009, result): “6] [AM07], and P. Zhang and Z. Zhang [ZZ06]. We also mention that for the full system (E) there are well posedness results given by D. Christodoulou and H. Lindblad [CL00], D. Coutand and S. Shkoller [CS05], J. Shatah and C. Zeng [SZ06]. Recently S. J. Wu proved almost global existence for small dat”
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].”
Global solutions and asymptotic behavior for two dimensional gravity water waves (2013, methodology): “Several extensions of their results have been obtained and we refer the reader to Córdoba, Córdoba and Gancedo [20], CoutandShkoller [22], Lannes [48, 49, 47], Linblad [50], Masmoudi-Rousset [52] and Shatah-Zeng [57, 58] for recent results on the Cauchy problem for the gravity water waves equation”
Well‐posedness in smooth function spaces for moving‐boundary 1‐D compressible euler equations in physical vacuum (2009, methodology): “We combine the methodology of our a priori estimates [2], with a particular degenerate parabolic regularization of Euler, which follows our methodology in [3, 4], as well as a new higher-order Hardy-type inequality which permits”
Well-posedness of the Muskat problem with H2 initial data (2014, positive): “Our method can also be applied to the Stefan problem [39], to the free-boundary problem for the incompressible Euler equations [23, 26], as well as to the compressible Euler equations [21, 25] .”