Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients

Abstract

The linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows is studied. As in our earlier work (Chen et al. in Adv Math 311:18–60, 2017b) on the linear stability with constant coefficients, the problem has a free boundary which is characteristic, and also the Kreiss–Lopatinskii condition is not uniformly satisfied. In addition, the roots of the Lopatinskii determinant of the para-linearized system may coincide with the poles of the system. Such a new collapsing phenomenon causes serious difficulties when applying the bicharacteristic extension method of Coulombel (SIAM J Math Anal 34(1):142–172, 2002; Ann Inst H Poincaré Anal Non Linéaire 21(4):401–443, 2004) and Coulombel and Secchi (Indiana Univ Math J 53(4):941–1012, 2004). Motivated by our method introduced in the constant-coefficient case (Adv Math 311:18–60, 2017b), we perform an upper triangularization to the para-linearized system to separate the outgoing mode into a closed form where the outgoing mode only appears at the leading order. This procedure results in a gain of regularity for the outgoing mode, which allows us to overcome the loss of regularity of the characteristic components at the poles and hence to close all the energy estimates. We find that, analogous to the constant-coefficient case, elasticity generates notable stabilization effects, and there are additional stable subsonic regions compared with the isentropic Euler flows. Moreover, since our method does not rely on the construction of the characterisic curves, it can also be applied to other fluid models such as the non-isentropic Euler equations and the MHD equations.