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Nonlinear spectral instability of steady-state flow of a viscous liquid past a rotating obstacle

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Abstract

We show that a steady-state solution to the system of equations of a Navier–Stokes flow past a rotating body is nonlinearly unstable if the associated linear operator \(\mathcal {L}\) has a part of the spectrum in the half-plane \(\{\lambda \in \mathbb {C};\ \mathrm{Re}\, \lambda >0\}\). Our result does not follow from known methods, mainly because the basic nonlinear operator is not bounded in the same space in which the instability is studied. As an auxiliary result of independent interest, we also show that the uniform growth bound of the \(C_0\)-semigroup \(\mathrm {e}^{\mathcal {L}t}\) is equal to the spectral bound of operator \(\mathcal {L}\).

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Acknowledgements

Part of this work was carried out when the first author was tenured with the Eduard Čech Distinguished Professorship at the Mathematical Institute of the Czech Academy of Sciences in Prague. His work is also partially supported by NSF Grant DMS-1614011 and the Mathematical Institute of the Czech Academy of Sciences (RVO 67985840). The second author also acknowledges the support of the Grant Agency of the Czech Republic (Grant no. 17-01747S).

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, USA

    Giovanni P. Galdi

  2. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67, Prague 1, Czech Republic

    Jiří Neustupa

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  1. Giovanni P. Galdi
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  2. Jiří Neustupa
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Correspondence to Jiří Neustupa.

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Communicated by Y. Giga.

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Galdi, G.P., Neustupa, J. Nonlinear spectral instability of steady-state flow of a viscous liquid past a rotating obstacle. Math. Ann. 382, 357–382 (2022). https://doi.org/10.1007/s00208-020-02045-x

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