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We consider the stability problem for shock layers in Slemrod's model of anisentropic gas with capillarity. We show that these traveling waves aremonotone in the weak capillarity case, and become highly oscillatory as thecapillarity strength increases. Using a spectral energy estimate we prove thatsmall-amplitude monotone shocks are spectrally stable. We also show, throughthe use of a novel spectral energy estimate, that monotone shocks have nounstable real spectrum regardless of amplitude; this implies that anyinstabilities of these monotone traveling waves, if they exist, must occurthrough a Hopf-like bifurcation, where one or more conjugate pairs ofeigenvalues cross the imaginary axis. We then conduct a systematic numericalEvans function study, which shows that monotone and mildly oscillatory profilesin an adiabatic gas are spectrally stable for moderate values of shock andcapillarity strengths. In particular, we show that the transition from monotoneto non-monotone profiles does not appear to trigger any instabilities.

On the shock wave spectrum for isentropic gas dynamics with capillarity