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Linear Stability of Steady Flows of Jeffreys Type Fluids

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Recent Trends in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 35))

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Abstract

We establish a rigorous criterion for linear stability of a class of viscoelastic flows. The analysis is based on recent results for advective systems.

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References

  1. Blake, M.D.: A spectral bound for asymptotically norm-continuous semigroups. J. Operator Th. 45, 111–130 (2001)

    MathSciNet  MATH  Google Scholar 

  2. Friedlander, S., Vishik, M.M.: Instability criteria for the flow of an inviscid incompressible fluid. Phys. Rev. Lett. 66, 2204–2206 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hille, E., Phillips, R.: Functional Analysis and Semigroups, vol. 31. Am. Soc. Coll. Publ., Providence (1957)

    MATH  Google Scholar 

  4. Lifschitz, A., Hameiri, E.: Local stability conditions in fluid dynamics. Phys. Fluids A 3, 2644–2651 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lifschitz, A., Hameiri, E.: Localized instabilities of vortex rings with swirl. Commun. Pure Appl. Math. 46, 1379–1408 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Nagel, R., Poland, J.: The critical spectrum of a strongly continuous semigroup. Adv. Math. 152, 120–133 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Renardy, M.: On the linear stability of hyperbolic PDEs and viscoelastic flows. Z. angew. Math. Phys. 45, 854–865 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Renardy, M.: Stability of creeping flows of Maxwell fluids. Arch. Ration. Mech. Anal. 198, 723–733 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Renardy, M.: Nonlinear stability for advective systems. J. Evolution Eqns. 10, 955–963 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Renardy, M.: Stability of steady flows of multimode Maxwell fluids. J. Evolution Eqns. 11, 847–860 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Shvydkoy, R.: The essential spectrum of advective equations. Commun. Math. Phys. 265, 507–545 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shvydkoy, R.: Continuous spectrum of the 3d Euler equations is a solid annulus. C.R. Acad. Sci. Paris 348, 897–900 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Shvydkoy, R., Latushkin, Y.: Operator algebras and the Fredholm spectrum of advective equations of linear hydrodynamics. J. Functional Anal. 257, 3309–3328 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Vishik, M., Friedlander, S.: Dynamo theory methods for hydrodynamic stability. J. Math. Pures Appl. 72, 145–180 (1993)

    MathSciNet  MATH  Google Scholar 

  15. Zabczyk, J.: A note on C 0 semigroups. Bull. Acad. Polon. Sc. (Math. Astr. Phys.) 23, 895–898 (1975)

    Google Scholar 

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Acknowledgements

This research was supported by the National Science Foundation under Grant DMS-1008426.

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Correspondence to Michael Renardy .

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Renardy, M. (2013). Linear Stability of Steady Flows of Jeffreys Type Fluids. In: Johann, A., Kruse, HP., Rupp, F., Schmitz, S. (eds) Recent Trends in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 35. Springer, Basel. https://doi.org/10.1007/978-3-0348-0451-6_23

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