Abstract
Extending results of Humpherys–Lyng–Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier–Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients \({\mu}\), \({\mu +\eta}\), and \({\nu=\kappa/c_v}\) constant and in the physical ratios predicted by statistical mechanics, and Mach number \({M > 1.035}\). Our results indicate unconditional stability within the parameter range considered; this agrees with the results of Erpenbeck and Majda for the corresponding inviscid case of Euler shocks. Notably, this study includes the first successful numerical computation of an Evans function associated with the multidimensional stability of a viscous shock wave. The methods introduced can be used in principle to decide stability for shocks in any polytropic gas, or indeed for shocks of other models, including in, particular, viscoelasticity, combustion, and magnetohydrodynamics (MHD).
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Communicated by A. Bressan
J.H. was partially supported by NSF grant DMS-CAREER-0847074. G.L. was partially supported by NSF grants DMS-CAREER-0845127 and DMS-1413273. K.Z. was partially supported by NSF grants DMS-0300487 and DMS-0801745.
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Humpherys, J., Lyng, G. & Zumbrun, K. Multidimensional Stability of Large-Amplitude Navier–Stokes Shocks. Arch Rational Mech Anal 226, 923–973 (2017). https://doi.org/10.1007/s00205-017-1147-7
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DOI: https://doi.org/10.1007/s00205-017-1147-7