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Dynamics near Unstable, Interfacial Fluids

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Abstract

We study three examples of unstable interfacial fluid motions: vortex sheets with surface tension, Hele-Shaw flows with surface tension, and vortex patches. In all three cases, the nonlinear dynamics of a large class of smooth perturbations is proven to be characterized by the corresponding fastest linear growing mode(s) up to the time scale of \(log \frac{1}{\delta }\), where \(\delta \) is the magnitude of the initial perturbation. In all three cases, the analysis is based on an unified analytical framework that includes precise bounds on the growth of the linearized operator, given by an explicit solution formula, as well as a special sharp nonlinear energy growth estimate. Our main contribution is establishing this nonlinear energy growth estimate for each interface problem in certain high energy norms.

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Author information

Authors and Affiliations

  1. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    Yan Guo

  2. Department of Mathematics, University of Portland, Portland, OR, 97203, USA

    Chris Hallstrom

  3. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Daniel Spirn

Authors
  1. Yan Guo
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  2. Chris Hallstrom
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  3. Daniel Spirn
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Corresponding author

Correspondence to Chris Hallstrom.

Additional information

Communicated by P. Constantin

Y. G. was supported in part by NSF grants DMS-0305161, INT-9815432 and a Salomon award of Brown University.

D. S. was supported in part by NSF grant DMS-0510121.

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Guo, Y., Hallstrom, C. & Spirn, D. Dynamics near Unstable, Interfacial Fluids. Commun. Math. Phys. 270, 635–689 (2007). https://doi.org/10.1007/s00220-006-0164-4

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  • DOI: https://doi.org/10.1007/s00220-006-0164-4

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