Global solutions for the gravity water waves equation in dimension 3

By Pierre Germain, Nader Masmoudi, Jalal Shatah. 2009

384 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ec63d76a11f5358424b423630f75b7b053566cd5

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In Mathematical Physics

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Citations in the corpus

Global solutions and asymptotic behavior for two dimensional gravity water waves (2013, background): “Germain–Masmoudi–Shatah recently proved global existence for pure capillary waves in dimension d = 2 in [34].”
On the regularization mechanism for the periodic Korteweg–de Vries equation (2009, background): “Furthermore, different nonlinear dispersive and wave equations, such as the m-KdV, the Klein-Gordon equation and water wave equations (see for example, [13], [19], [23]-[25], and references therein), can possibly be studied quite similarly, a subject of future research.”
MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS (2013, result): “We also note that although our approach is close in spirit to recent developments in global existence for quasilinear equations [43, 44, 63, 64, 65], some of the key estimates really pertain to the low-regularity theory (see Lemma 7.”
Local well-posedness of the viscous surface wave problem without surface tension (2013, methodology): “Local well-posedness in this framework was established by Wu [31, 32] and Lannes [22], an almost-global well-posedness result was then proved by Wu [33] for the 2D problem, and global well-posedness was proved by Wu [34] and Germain-Masmoudi-Shatah [17] in 3D.”
Well‐posedness of Compressible Euler Equations in a Physical Vacuum (2010, background): “For multi-dimensional flows, parallel to a lot of activity and important progress in free surface boundary problems, in particular regarding incompressible fluids [2, 3, 4, 11, 20, 35, 58, 61, 64], more recently, there are some works trying to prove the local well-posedness of physical vacuum in 3D.”
Finite time singularities for the free boundary incompressible Euler equations (2011, background): “[19], [20] and Wu [33] proved existence for all time in the case of small initial data.”
Almost Exponential Decay of Periodic Viscous Surface Waves without Surface Tension (2013, background): “2), both with and without viscosity and surface tension: [1, 2, 3, 4, 5, 6, 7, 8, 11, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30].”
Almost global wellposedness of the 2-D full water wave problem (2009, background): “Finally, we mention the recent work of Germain, Masmoudi, Shatah [13], in which they assumed smallness of (h, |D|1/2ψ) in L2(|x|2 dx) ∩ W 6,1(dx) ∩ HN(dx) for some large N initially and studied the global existence and scattering of the 3D water wave through analyzing the space-time resonance.”
On the Cauchy problem for gravity water waves (2012, methodology): “Several extensions of their results have been obtained by different methods (see [22, 31, 32, 33, 35, 41, 54, 55, 59] for recent results and the surveys [11, 29, 36] for more references).”
Global Well-Posedness of the Incompressible Magnetohydrodynamics (2016, methodology): “Germain, P.,Masmoudi, N., Shatah, J.: Global solutions for the gravity water waves equation in dimension 3.”
Global existence for the Euler-Maxwell system (2011, positive): “It was also used to prove global existence of small data solutions for water waves [13, 14].”
Decay of viscous surface waves without surface tension (2010, neutral): “2), both with and without viscosity and surface tension: [1, 3, 4, 5, 6, 7, 8, 9, 11, 15, 16, 17, 18, 21, 20, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32].”
Global small solutions to 2-D incompressible MHD system (2013, positive): “1) As in [11], [12], [13], [17], [27], [30], [32] and various earlier references therein, we shall use the Lagrangian formulation.”
Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations (2013, background): “Phase mixing also shares certain similarities with scattering in the theory of dispersive wave equations (see for instance [50, 34, 58]) as already pointed out in [27, 18].”
Time quasi-periodic gravity water waves in finite depth (2017, neutral): “For three dimensional fluids (i.e. d = 2) it has been proved independently by GermainMasmoudi-Shatah [31] and Wu [60].”