Abstract
In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier–Stokes equations, or p-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength.
By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there are no unstable spectra in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected.
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Communicated by P. Constantin
This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487.
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Barker, B., Humpherys, J., Rudd, K. et al. Stability of Viscous Shocks in Isentropic Gas Dynamics. Commun. Math. Phys. 281, 231–249 (2008). https://doi.org/10.1007/s00220-008-0487-4
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DOI: https://doi.org/10.1007/s00220-008-0487-4