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Linear Inviscid Damping for Monotone Shear Flows in a Finite Periodic Channel, Boundary Effects, Blow-up and Critical Sobolev Regularity

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Abstract

In a previous article (Zillinger, Linear inviscid damping for monotone shear flows, 2014), we have established linear inviscid damping for a large class of monotone shear flows in a finite periodic channel and have further shown that boundary effects asymptotically lead to the formation of singularities of derivatives of the solution as \({t \rightarrow \infty}\). As the main results of this article, we provide a detailed description of the singularity formation and establish stability in all sub-critical fractional Sobolev spaces and blow-up in all super-critical spaces. Furthermore, we discuss the implications of the blow-up to the problem of nonlinear inviscid damping in a finite periodic channel, where high regularity would be essential to control nonlinear effects.

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Authors and Affiliations

  1. Mathematisches Institut, Universität Bonn, 53115, Bonn, Germany

    Christian Zillinger

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  1. Christian Zillinger
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Correspondence to Christian Zillinger.

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Communicated by V. Šverák

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Zillinger, C. Linear Inviscid Damping for Monotone Shear Flows in a Finite Periodic Channel, Boundary Effects, Blow-up and Critical Sobolev Regularity. Arch Rational Mech Anal 221, 1449–1509 (2016). https://doi.org/10.1007/s00205-016-0991-1

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  • DOI: https://doi.org/10.1007/s00205-016-0991-1

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