On the stability of two-dimensional nonisentropic elastic vortex sheets

Abstract

<p style="text-indent:20px;">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<inline-formula><tex-math id="M1">\begin{document}$ \breve{{\mathrm{i}}} $\end{document}</tex-math></inline-formula> determinant extremely complicated. Our results are twofold. First, through a qualitative analysis of the roots of the Lopatinski<inline-formula><tex-math id="M2">\begin{document}$ \breve{{\mathrm{i}}} $\end{document}</tex-math></inline-formula> 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 [<xref ref-type="bibr" rid="b6">6</xref>] 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.</p>