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Stability of Isentropic Navier–Stokes Shocks in the High-Mach Number Limit

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Abstract

By a combination of asymptotic ODE estimates and numerical Evans function calculations, we establish stability of viscous shock solutions of the isentropic compressible Navier–Stokes equations with γ-law pressure (i) in the limit as Mach number M goes to infinity, for any γ ≥ 1 (proved analytically), and (ii) for M ≥ 2,500, \({\gamma\in [1,2.5]}\) or M ≥ 13,000, \({\gamma \in [2.5,3]}\) (demonstrated numerically). This builds on and completes earlier studies by Matsumura–Nishihara and Barker–Humpherys–Rudd–Zumbrun establishing stability for low and intermediate Mach numbers, respectively, indicating unconditional stability, independent of shock amplitude, of viscous shock waves for γ-law gas dynamics in the range \({\gamma \in [1,3]}\) . Other γ-values may be treated similarly, but have not been checked numerically. The main idea is to establish convergence of the Evans function in the high-Mach number limit to that of a pressureless, or “infinitely compressible”, gas with additional upstream boundary condition determined by a boundary-layer analysis. Recall that low-Mach number behavior is formally incompressible.

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Authors and Affiliations

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

    Jeffrey Humpherys

  2. LAGA, Institut Galilee, Universite Paris 13, 93 430, Villetaneuse, France

    Olivier Lafitte

  3. CEA Saclay, DM2S/DIR, 91 191, Gif sur Yvette Cedex, France

    Olivier Lafitte

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

    Kevin Zumbrun

Authors
  1. Jeffrey Humpherys
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  2. Olivier Lafitte
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  3. Kevin Zumbrun
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Correspondence to Kevin Zumbrun.

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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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Humpherys, J., Lafitte, O. & Zumbrun, K. Stability of Isentropic Navier–Stokes Shocks in the High-Mach Number Limit. Commun. Math. Phys. 293, 1–36 (2010). https://doi.org/10.1007/s00220-009-0885-2

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