Journal of Mathematical Fluid Mechanics

, Volume 20, Issue 3, pp 1195–1211 | Cite as

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

  • Roman ShvydkoyEmail author


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.


Rayleigh–Taylor instability Essential spectrum Pseudo-differential operator WKB 

Mathematics Subject Classification

76E20 35P05 47D06 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer Science, M/C 249University of IllinoisChicagoUSA

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