On a Hele-Shaw Type Domain Evolution with Convected Surface Energy Density: The Third-Order Problem

@article{Gnther2006OnAH,
  title={On a Hele-Shaw Type Domain Evolution with Convected Surface Energy Density: The Third-Order Problem},
  author={Matthias G{\"u}nther and Georg Prokert},
  journal={SIAM J. Math. Anal.},
  year={2006},
  volume={38},
  pages={1154-1185}
}
We investigate a moving boundary problem with a gradient flow structure which generalizes Hele-Shaw flow driven solely by surface tension to the case of nonconstant surface ten- sion coefficient taken along with the liquid particles at boundary. The resulting evolution problem is first order in time, contains a third-order nonlinear pseudodifferential operator and is degenerate parabolic. Well-posedness of this problem in Sobolev scales is proved. The main tool is the con- struction of a… 

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On a Hele-Shaw-Type Domain Evolution with Convected Surface Energy Density
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Well-posedness of the resulting evolution problem in Sobolev scales is proved, including cases in which the surface tension coefficient degenerates, and an abstract result on Galerkin approximations with variable bilinear forms is applied.
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This paper addresses short-time existence and uniqueness of a solution to the N-dimensional Hele–Shaw flow problem with surface tension as driving mechanism. Global existence in time and exponential
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Abstract.Short-time existence, uniqueness, and regularity results are shown for the moving boundary problem of a free drop of liquid governed by the Stokes equations and driven by surface tension.
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We provide numerical and analytic evidence for the formation of a singularity driven only by surface tension in the mathematical model describing a two‐dimensional Hele–Shaw cell with no air
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Abstract. It has been recently discovered that both the surface tension driven one-phase Hele-Shaw flow and its lubrication approximation can be understood as (continuous limits of time-discretized)
Local and global existence results for anisotropic Hele-Shaw flows
In this paper we study a moving boundary problem for an anisotropic two-phase Hele–Shaw flow. Using a regularization technique, we prove existence of a local solution. Under suitable conditions on
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The main results of this paper are existence theorems for traveling gravity and cap- illary gravity water waves in two dimensions, and capillary gravity water waves in three dimensions, for any
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The Cauchy problem for the motion of a liquid drop under surface tension is solved locally in time on the basis of a general abstract existence theorem for Hamiltonian systems which seems to be of
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The surfactant influence on the bubble motion in a Hele–Shaw cell was studied experimentally. In order to differentiate the cases with and without the surfactant influence, the motion of air bubbles
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