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Spectral Stability of Ideal-Gas Shock Layers

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Abstract

Extending recent results in the isentropic case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the spectral stability of shock-wave solutions of the compressible Navier–Stokes equations with ideal gas equation of state. Our main results are that, in appropriately rescaled coordinates, the Evans function associated with the linearized operator about the wave (i) converges in the large-amplitude limit to the Evans function for a limiting shock profile of the same equations, for which internal energy vanishes at one end state; and (ii) has no unstable (positive real part) zeros outside a uniform ball |λ| ≦ Λ. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with the (nonphysical) limiting case, form a compact family of boundary-value problems that can be conveniently investigated numerically. An extensive numerical Evans-function study yields one-dimensional spectral stability, independent of amplitude, for gas constant γ in [1.2, 3] and ratio ν/μ of heat conduction to viscosity coefficient within [0.2, 5] (γ ≈ 1.4, ν/μ ≈ 1.47 for air). Other values may be treated similarly but were not considered. The method of analysis extends also to the multi-dimensional case, a direction that we shall pursue in a future work.

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Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Jeffrey Humpherys

  2. Department of Mathematics, University of Wyoming, Laramie, WY, 82071, USA

    Gregory Lyng

  3. Department of Mathematics, Indiana University, Bloomington, IN, 47402, USA

    Kevin Zumbrun

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  1. Jeffrey Humpherys
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  2. Gregory Lyng
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  3. Kevin Zumbrun
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Correspondence to Jeffrey Humpherys.

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Communicated by C.M. Dafermos

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Humpherys, J., Lyng, G. & Zumbrun, K. Spectral Stability of Ideal-Gas Shock Layers. Arch Rational Mech Anal 194, 1029–1079 (2009). https://doi.org/10.1007/s00205-008-0195-4

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