Geometry and a priori estimates for free boundary problems of the Euler's equation

By Jalal Shatah, C. Zeng. 2006

309 citations recorded by the source corpus.

In this paper we derive estimates to the free boundary problem for the Euler equation with surface tension, and without surface tension provided the Rayleigh‐Taylor sign condition holds. We prove that as the surface tension tends to 0, when the Rayleigh‐Taylor condition is satisfied, solutions converge to the Euler flow with zero surface tension. © 2007 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Mechanics

View Paper: https://www.semanticscholar.org/paper/38b862ba507c8023375a03f464b4d3393ff2ee9d

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In Applied Mathematics

In Computational Mechanics

Citations in the corpus

On the Cauchy problem for gravity water waves (2012, methodology): “ In fact, the systematic use of the Lagrangian formulation in most previous works [9, 54, 55], and the intensive use of Riemannian geometry tools (parallel transport, vector ﬁelds,...) by Shatah-Zeng [49, 50, 51], Christodoulou–Lindblad [21] or Lindblad [41] seem to at least require bounded curvatur”
Decay of viscous surface waves without surface tension (2010, unknown): “22”
Almost Exponential Decay of Periodic Viscous Surface Waves without Surface Tension (2013, unknown): “Ē−2N (0) + ∫ t 0 Eθ2ND2N + εD2N + ε−8N−1D̄02N .”
Large time existence for 3D water-waves and asymptotics (2007, background): “P regime (with ﬂat bottom), one takes P = {(ε,ε, √ ε),ε∈ (0,1)} ×{0}, etc. Remark 5.3. The condition infR2 a(ζ p 0,ψ0) >0 is the classical Taylor sign condition proper to the water-wave equations ([47,48,29,33, 34,3,12,44], among others). It is obviously true for small data and we give in Proposi”
Well-posedness of the free-surface incompressible Euler equations with or without surface tension (2005, unknown): “…g (η) on Γ × (0, T ] , (1.3d) (η, v) = (Id, u 0 ) on Ω × {t = 0} , (1.3e) where N denotes the unit normal to Γ, and ∆ g is the surface Laplacian with respect to the induced metric g on Γ, written in local coordinates as ∆ g = |g| −1 ∂ α [ |g|g αβ ∂ β ] , g αβ = [g αβ ] −1 , g αβ = η ,α · η ,β ,…”
Local well-posedness of the viscous surface wave problem without surface tension (2013, methodology): “Local well-posedness without the irrotationality assumption was proved with a modified surface formulation by Zhang and Zhang [2008] and with the original formulation in [[Christodoulou and Lindblad 2000; Lindblad 2005; Coutand and Shkoller 2007; Shatah and Zeng 2008]].”
Global wellposedness of the 3-D full water wave problem (2011, background): “We mention the following recent work on local wellposedness [1, 5, 6, 15, 18, 19, 21, 24, 34].”
Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations (2012, background): “mption) there are well posedness results starting with the works by Christodoulou and Lindblad [12] and Lindblad [34]. Alternative approach were proposed by Coutand and Shkoller [13], Shatah and Zeng [49] and P. Zhang and Z. Zhang [61]. Note that the well-posedness of the Euler equation requires a T”
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].”
On the water-wave equations with surface tension (2009, background): “The purpose of this article is twofold: first we want to clarify the Cauchy theory as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom (to our knowledge, previous results required the bottom to be the graph of an H13 function).”
Finite time singularities for the free boundary incompressible Euler equations (2011, background): “s non-self-intersecting (satises the arc-chord condition). For recent work on local existence see Wu [28], Christodoulou-Lindblad [12], Lindblad [21], Lannes [19], Coutand-Shkoller [15], Shatah-Zeng [25], Zhang-Zhang [32], AlazardMetivier [2], Ambrose-Masmoudi [4], Cordoba et al. [13], Lannes [20], ”
Well‐posedness in smooth function spaces for moving‐boundary 1‐D compressible euler equations in physical vacuum (2009, background): “tive loss in linearized equations when arbitrary domains are considered, or complex analysis tools for the irrotational problem with inﬁnite depth. We refer the reader to [2], [10], [20], [22], [33], [35], [41], [42], [43], and [46] for a partial list of papers on this topic. 6 D. COUTAND AND S. SHK”
Local Well-Posedness for Fluid Interface Problems (2011, methodology): “Finally, while the estimates on the vorticity, which can be obtained easily, along with vir are sufficient for one to obtain the estimates of the velocity in [ 23 ,24], they are not sufficient to reconstruct the velocity if the domains are not simply connected.”
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].”
Global solutions for the gravity water waves equation in dimension 3 (2009, methodology): “on S. This is done by standard estimate on harmonic functions on Ω which imply that as far as estimates are concerned, N behaves like √ −∆S plus lower order terms. These estimates were carried out in [SZ06] without detailing explicitly how the constants in the inequalities depend on the various norm”