# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021083

## On the stability of two-dimensional nonisentropic elastic vortex sheets

 1 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA 2 Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 3 School of Mathematical Sciences, Beijing Normal University and Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

* Corresponding author

Dedicated to Professor Shuxing Chen on the Occasion of His 80th Birthday

Received  February 2021 Revised  April 2021 Published  May 2021

Fund Project: R. M. Chen is supported in part by the NSF grant DMS-1907584. F. Huang was supported in part by National Center for Mathematics and Interdisciplinary Sciences, AMSS, CAS and NSFC Grant No. 11371349 and 11688101. D. Wang was supported in part by NSF grant DMS-1907519. D. Yuan was supported by China Scholarship Council No.201704910503, NSFC Grant No.12001045 and China Postdoctoral Science Foundation No.2020M680428

We are concerned with the stability of vortex sheet solutions for the two-dimensional nonisentropic compressible flows in elastodynamics. This is a nonlinear free boundary hyperbolic problem with characteristic discontinuities, which has extra difficulties when considering the effect of entropy. The addition of the thermal effect to the system makes the analysis of the Lopatinski$\breve{{\mathrm{i}}}$ determinant extremely complicated. Our results are twofold. First, through a qualitative analysis of the roots of the Lopatinski$\breve{{\mathrm{i}}}$ determinant for the linearized problem, we find that the vortex sheets are weakly stable in some supersonic and subsonic regions. Second, under the small perturbation of entropy, the nonlinear stability can be adapted from the previous two-dimensional isentropic elastic vortex sheets [6] by applying the Nash-Moser iteration. The two results confirm the strong elastic stabilization of the vortex sheets. In particular, our conditions for the linear stability (1) ensure that a stable supersonic regime as well as a stable subsonic one always persist for any given nonisentropic configuration, and (2) show how the stability condition changes with the thermal fluctuation. The existence of the stable subsonic bubble, a phenomenon not observed in the Euler flow, is specially due to elasticity.

Citation: Robin Ming Chen, Feimin Huang, Dehua Wang, Difan Yuan. On the stability of two-dimensional nonisentropic elastic vortex sheets. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021083
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