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The linear stability with variable coefficients of the vortex sheets for thetwo-dimensional compressible elastic flows is studied. As in our earlier workon the linear stability with constant coefficients, the problem has a freeboundary which is characteristic, and also the Kreiss-Lopatinskii condition isnot uniformly satisfied. In addition, the roots of the Lopatinskii determinantof the para-linearized system may coincide with the poles of the system. Such anew collapsing phenomenon causes serious difficulties when applying thebicharacteristic extension method. Motivated by our method introduced in theconstant-coefficient case, we perform an upper triangularization to thepara-linearized system to separate the outgoing mode into a closed form wherethe outgoing mode only appears at the leading order. This procedure results ina gain of regularity for the outgoing mode which allows us to overcome the lossof regularity of the characteristic components at the poles, and hence to closeall the energy estimates. We find that, analogous to the constant coefficientcase, elasticity generates notable stabilization effects, and there areadditional stable subsonic regions compared with the isentropic Euler flows.Moreover, since our method does not rely on the construction of thebicharacterisic curves, it can also be applied to other fluid models such asthe non-isentropic Euler equations and the MHD equations.

Linear Stability of Compressible Vortex Sheets in 2D Elastodynamics: Variable Coefficients