On the Cauchy problem for a capillary drop. Part I: irrotational motion

Beyer, Klaus; Günther, Matthias

doi:10.1002/(sici)1099-1476(199808)21:12<1149::aid-mma990>3.0.co;2-c

Cited by 86 publications

(81 citation statements)

References 13 publications

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“…In the framework of Sobolev spaces and without smallness assumptions on the initial data, the local wellposedness was first obtained by Wu [17,18] for the gravity waves system. The local wellposedness was also obtained when the surface tension effect is effective by Beyer-Günther [7]. Later, different methods were developed and many important results were obtained to improve our understanding on the local behavior of the water waves system.…”

Section: Previous Resultsmentioning

confidence: 99%

On the 3-dimensional water waves system above a flat bottom

Wang¹

2017

Analysis & PDE

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As a starting point of studying the long time behavior of the 3D water waves system in the flat bottom setting, in this paper, we try to improve the understanding of the Dirichlet-Neumann operator in this setting. As an application, we study the 3D gravity waves system and derive a new L 2 − L ∞ type energy estimate, which has a good structure in the L ∞ −type space. In our second paper [16], base on the results we obtained in this paper, we prove the global regularity of the 3D gravity waves system for suitably small initial data.

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Section: Previous Resultsmentioning

confidence: 99%

On the 3-dimensional water waves system above a flat bottom

Wang¹

2017

Analysis & PDE

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“…Many results have been obtained on the Cauchy theory for the water-waves system, starting from the pioneering works of Nalimov [51], Yoshihara [70], Craig [29] (see also Hou, Teng and Zhang [36] and Beale, Hou and Lowengrub [14]). In the framework of Sobolev spaces and without smallness assumptions on the data, the well-posedness of the Cauchy problem was first proved by Beyer-Günther in [15] in the case with surface tension (in any number of space dimensions) and by Wu for the case without surface tension (see [67] for 2D water waves and [68] for the general case d ≥ 1). Several extensions of their results have been obtained by different methods.…”

Section: Known Resultsmentioning

confidence: 99%

Low regularity Cauchy theory for the water-waves problem: canals and swimming pools

Alazard¹,

Burq²,

Zuily³

2012

Journées Équations Aux Dérivées Partielles

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“…We note a result on nonlinear interpolation, whose proof can be found in [2], Propostion A.1 and Remark A.2.…”

Section: This Immediately Implies (511)mentioning

confidence: 90%

On a Hele--Shaw-Type Domain Evolution with Convected Surface Energy Density

Günther

Prokert²

2005

SIAM J. Math. Anal.

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Interest is directed to a moving boundary problem with a gradient flow structure which generalizes surface-tension driven Hele-Shaw flow to the case of nonconstant surface tension coefficient taken along with the liquid particles at the boundary. In the case with kinetic undercooling regularization well-posedness of the resulting evolution problem in Sobolev scales is proved, including cases in which the surface tension coefficient degenerates. The problem is reformulated as a vector-valued, degenerate parabolic Cauchy problem. To solve this, we prove and apply an abstract result on Galerkin approximations with variable bilinear forms.

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On the Cauchy problem for a capillary drop. Part I: irrotational motion

Cited by 86 publications

References 13 publications

On the 3-dimensional water waves system above a flat bottom

On the 3-dimensional water waves system above a flat bottom

Low regularity Cauchy theory for the water-waves problem: canals and swimming pools

On a Hele--Shaw-Type Domain Evolution with Convected Surface Energy Density

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