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On the Existence for the Free Interface 2D Euler Equation with a Localized Vorticity Condition

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Abstract

We prove a local-in-time existence of solutions result for the two dimensional incompressible Euler equations with a moving boundary, with no surface tension, under the Rayleigh–Taylor stability condition. The main feature of the result is a local regularity assumption on the initial vorticity, namely \(H^{1.5+\delta }\) Sobolev regularity in the vicinity of the moving interface in addition to the global regularity assumption on the initial fluid velocity in the \(H^{2+\delta }\) space. We use a special change of variables and derive a priori estimates, establishing the local-in-time existence in \(H^{2+\delta }\). The assumptions on the initial data constitute the minimal set of assumptions for the existence of solutions to the rotational flow problem to be established in 2D.

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Acknowledgments

I.K. and F.W. were supported in part by the NSF Grant DMS-1311943, while V.V. was supported in part by the NSF Grant DMS-1514771 and an A.P. Sloan fellowship.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Igor Kukavica & Fei Wang

  2. Department of Mathematics, American University of Sharjah, Sharjah, UAE

    Amjad Tuffaha

  3. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Vlad Vicol

Authors
  1. Igor Kukavica
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  2. Amjad Tuffaha
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  3. Vlad Vicol
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  4. Fei Wang
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Correspondence to Igor Kukavica.

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To the memory of Professor A.V. Balakrishnan.

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Kukavica, I., Tuffaha, A., Vicol, V. et al. On the Existence for the Free Interface 2D Euler Equation with a Localized Vorticity Condition. Appl Math Optim 73, 523–544 (2016). https://doi.org/10.1007/s00245-016-9346-4

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