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On the Vortex Sheets of Compressible Flows

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Abstract

This paper provides a review of the recent results on the stability of vortex sheets in compressible flows. Vortex sheets are contact discontinuities of the underlying flows. The vortex sheet problem is a free boundary problem with a characteristic boundary and is challenging in analysis. The formulation of the vortex sheet problem will be introduced. The linear stability and nonlinear stability for both the two-dimensional two-phase compressible flows and the two-dimensional elastic flows are summarized. The linear stability of vortex sheets for the three-dimensional elastic flows is also presented. The difficulties of the vortex sheet problems and the ideas of proofs are discussed.

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Acknowledgements

R. M. Chen is supported in part by the NSF grant DMS-1907584. F. Huang was supported in part by the National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and the National Natural Sciences Foundation of China under Grant Nos. 11371349 and 11688101. D. Wang was supported in part by the NSF under grant DMS-1907519. D. Yuan was supported in part by the National Natural Sciences Foundation of China under Grant No. 12001045 and the China Postdoctoral Science Foundation under Grant Nos. 2020M680428 and 2021T140063.

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Robin Ming Chen & Dehua Wang

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Feimin Huang

  3. Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Feimin Huang

  4. School of Mathematical Sciences, Beijing Normal University and Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

    Difan Yuan

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  1. Robin Ming Chen
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  2. Feimin Huang
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  3. Dehua Wang
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  4. Difan Yuan
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Correspondence to Dehua Wang.

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Chen, R.M., Huang, F., Wang, D. et al. On the Vortex Sheets of Compressible Flows. Commun. Appl. Math. Comput. 5, 967–986 (2023). https://doi.org/10.1007/s42967-022-00191-4

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  • DOI: https://doi.org/10.1007/s42967-022-00191-4

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