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On the Existence and Stability of 2D Compressible Current-Vortex Sheets

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Nonlinear Differential Equations and Applications (PICNDEA 2022)

Part of the book series: CIM Series in Mathematical Sciences ((CIMSMS,volume 7))

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Abstract

We study the existence and the stability of two-dimensional current-vortex sheets in ideal compressible magnetohydrodynamics. Under a suitable stability condition for the background state, we show that the linearized current-vortex sheets problem obeys an energy estimate in anisotropic weighted Sobolev spaces with a loss of derivatives. Then we establish the local-in-time existence and nonlinear stability of current-vortex sheets by a suitable Nash–Moser iteration, provided the stability condition is satisfied at each point of the initial discontinuity.

This survey paper presents our recent results in Morando et al. (Arch. Rational Mech. Anal. 247, 50 (2023)).

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Acknowledgements

The authors are grateful to the anonymous referee for her/his comments and suggestions that have contributed to improve this paper.

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Authors and Affiliations

  1. INdAM Unit & Department Civil, Environmental, Architectural Engineering and Mathematics (DICATAM), University of Brescia, Brescia, Italy

    Alessandro Morando, Paolo Secchi & Paola Trebeschi

  2. School of Mathematical Sciences, Bei**g Normal University and Laboratory of Mathematics and Complex Systems, Ministry of Education, Bei**g, China

    Difan Yuan

Authors
  1. Alessandro Morando
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  2. Paolo Secchi
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  3. Paola Trebeschi
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  4. Difan Yuan
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Corresponding author

Correspondence to Alessandro Morando .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Hugo Beirão da Veiga

  2. Department of Mathematics, Centro de Investigação em Matemática e Aplicações (CIMA), University of Évora, Évora, Portugal

    Feliz Minhós

  3. Department of Mathematics, Faculty of Sciences, University of Lisbon, Lisbon, Portugal

    Nicolas Van Goethem

  4. Department of Mathematics, Faculty of Sciences, University of Lisbon, Lisbon, Portugal

    Luís Sanchez Rodrigues

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Morando, A., Secchi, P., Trebeschi, P., Yuan, D. (2024). On the Existence and Stability of 2D Compressible Current-Vortex Sheets. In: Beirão da Veiga, H., Minhós, F., Van Goethem, N., Sanchez Rodrigues, L. (eds) Nonlinear Differential Equations and Applications. PICNDEA 2022. CIM Series in Mathematical Sciences, vol 7. Springer, Cham. https://doi.org/10.1007/978-3-031-53740-0_10

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