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Spectral Stability of Noncharacteristic Isentropic Navier–Stokes Boundary Layers

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Abstract

Building on the work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or shock-like, boundary layers of the isentropic compressible Navier–Stokes equations with γ-law pressure by a combination of asymptotic ODE estimates and numerical Evans function computations. Our analytical results include convergence of the Evans function in the shock and large-amplitude limits and stability in the large-amplitude limit, the first rigorous stability result for other than the nearly constant case, for all \({\gamma\geqq 1}\). Together with these analytical results, our numerical investigations indicate stability for γ ϵ [1, 3] for all compressive boundary-layers, independent of amplitude, save for inflow layers in the characteristic limit (not treated). Expansive inflow boundary-layers have been shown to be stable for all amplitudes by Matsumura and Nishihara using energy estimates. Besides the parameter of amplitude appearing in the shock case, the boundary-layer case features an additional parameter measuring displacement of the background profile, which greatly complicates the resulting case structure. Inflow boundary layers turn out to have quite delicate stability in both large-displacement (shock) and large-amplitude limits, necessitating the additional use of a mod-two stability index studied earlier by Serre and Zumbrun in order to decide stability.

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Authors and Affiliations

  1. Department of Mathematics, Penn. State University, University Park, PA, 16802, USA

    Nicola Costanzino

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

    Jeffrey Humpherys

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

    Toan Nguyen & Kevin Zumbrun

Authors
  1. Nicola Costanzino
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  2. Jeffrey Humpherys
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  3. Toan Nguyen
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  4. Kevin Zumbrun
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Correspondence to Kevin Zumbrun.

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Communicated by A. Bressan

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Costanzino, N., Humpherys, J., Nguyen, T. et al. Spectral Stability of Noncharacteristic Isentropic Navier–Stokes Boundary Layers. Arch Rational Mech Anal 192, 537–587 (2009). https://doi.org/10.1007/s00205-008-0153-1

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  • DOI: https://doi.org/10.1007/s00205-008-0153-1

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