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On the Rayleigh–Taylor Instability in Presence of a Background Shear

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Abstract

In this note we revisit the classical subject of the Rayleigh–Taylor instability in presence of an incompressible background shear flow. We derive a formula for the essential spectral radius of the evolution group generated by the linearization near the steady state and reveal that the velocity variations neutralize shortwave instabilities. The formula is a direct generalization of the result of Hwang and Guo (Arch Ration Mech Anal 167(3):235–253, (2003). Furthermore, we construct a class of steady states which posses unstable discrete spectrum with neutral essential spectrum. The technique involves the WKB analysis of the evolution equation and contains novel compactness criterion for pseudo-differential operators on unbounded domains.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, M/C 249, University of Illinois, Chicago, IL, 60607, USA

    Roman Shvydkoy

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  1. Roman Shvydkoy
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Correspondence to Roman Shvydkoy.

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The author thanks Zhiwu Lin and Chongchung Zeng for many fruitful conversations and Georgia Institute of Technology for hospitality. This research was partially supported by NSF grant DMS 1515705 and the College of LAS, UIC.

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Shvydkoy, R. On the Rayleigh–Taylor Instability in Presence of a Background Shear. J. Math. Fluid Mech. 20, 1195–1211 (2018). https://doi.org/10.1007/s00021-018-0362-9

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  • DOI: https://doi.org/10.1007/s00021-018-0362-9

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