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Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime

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Abstract

We are concerned with the nonlinear stability of vortex sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. This is a nonlinear hyperbolic problem with a characteristic free boundary. In this paper, we introduce a new symmetrization by choosing appropriate functions as primary unknowns. A necessary and sufficient condition for the weakly linear stability of relativistic vortex sheets is obtained by analyzing the roots of the Lopatinskiĭ determinant associated to the constant coefficient linearized problem. Under this stability condition, we show that the variable coefficient linearized problem obeys an energy estimate with a loss of derivatives. The construction of certain weight functions plays a crucial role in absorbing the error terms caused by microlocalization. Based on the weakly linear stability result, we establish the existence and nonlinear stability of relativistic vortex sheets under small initial perturbations by a Nash–Moser iteration scheme.

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Acknowledgments

The research of Gui-Qiang G. Chen was supported in part by the UK Engineering and Physical Sciences Research Council Award EP/E035027/1 and EP/L015811/1, and the Royal Society–Wolfson Research Merit Award (UK). The research of Paolo Secchi was supported in part by the Italian research projects PRIN 2012L5WXHJ-004 and PRIN 2015YCJY3A-004. The research of Tao Wang was supported in part by NSFC Grants #11601398 and #11731008, and the Italian research project PRIN 2012L5WXHJ-004. Tao Wang warmly thanks Prof. Alessandro Morando, Prof. Paolo Secchi, and Prof. Paola Trebeschi for support and hospitality during his postdoctoral stay at University of Brescia, and also expresses much gratitude to Prof. Huijiang Zhao for his continuous encouragement and constant support.

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Authors and Affiliations

  1. Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

    Gui-Qiang G. Chen

  2. AMSS & UCAS, Chinese Academy of Sciences, Beijing, 100190, China

    Gui-Qiang G. Chen

  3. DICATAM, Mathematical Division, University of Brescia, 25133, Brescia, Italy

    Paolo Secchi

  4. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Tao Wang

  5. DICATAM, Mathematical Division, University of Brescia, 25133, Brescia, Italy

    Tao Wang

  6. Hubei Key Laboratory of Computational Science, Wuhan University, Wuhan, 430072, China

    Tao Wang

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  1. Gui-Qiang G. Chen
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Correspondence to Tao Wang.

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Chen, GQ.G., Secchi, P. & Wang, T. Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime. Arch Rational Mech Anal 232, 591–695 (2019). https://doi.org/10.1007/s00205-018-1330-5

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