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Asymptotic Behavior of Solutions to the Compressible Navier–Stokes Equation Around a Parallel Flow

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Abstract

The initial boundary value problem for the compressible Navier–Stokes equation is considered in an infinite layer of \({\mathbf{R}^{2}}\) . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier–Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura–Nishida energy method.

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Authors and Affiliations

  1. Faculty of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan

    Yoshiyuki Kagei

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  1. Yoshiyuki Kagei
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Correspondence to Yoshiyuki Kagei.

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Communicated by T.-P. Liu

Dedicated to Professor Akitaka Matsumura on his 60th birthday

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Kagei, Y. Asymptotic Behavior of Solutions to the Compressible Navier–Stokes Equation Around a Parallel Flow. Arch Rational Mech Anal 205, 585–650 (2012). https://doi.org/10.1007/s00205-012-0516-5

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  • DOI: https://doi.org/10.1007/s00205-012-0516-5

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