Abstract:
We give an explicit construction of approximate eigenfunctions for a linearized Euler operator in dimensions two and three with periodic boundary conditions, and an estimate from below for its spectral bound in terms of an appropriate Lyapunov exponent. As a consequence, we prove that in dimension 2 the spectral and growth bounds for the corresponding group are equal. Therefore, the linear hydrodynamic stability of a steady state for the Euler equations in dimension 2 is equivalent to the fact that the spectrum of the linearized operator is pure imaginary. In dimension 3 we prove the estimate from below for the spectral bound that implies the same equality for every example where the relevant Lyapunov exponents could be effectively computed. For the kinematic dynamo operator describing the evolution of a magnetic field in an ideally conducting incompressible fluid we prove that the growth bound equals the spectral bound in dimensions 2 and 3.
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Received: 20 May 2002 / Accepted: 5 September 2002 Published online: 10 January 2003
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ID="*" The first author was partially supported by the Twinning Program of the National Academy of Sciences and National Science Foundation, and by the Research Council and Research Board of the University of Missouri.
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ID="**" The second author was partially supported by the National Science Foundation grant DMS 9876947 and CRDF grant RM1-2084.
Acknowledgements. The authors thank Susan Friedlander for useful discussions.
Communicated by P. Constantin
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Latushkin, Y., Vishik, M. Linear Stability in an Ideal Incompressible Fluid. Commun. Math. Phys. 233, 439–461 (2003). https://doi.org/10.1007/s00220-002-0775-3
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DOI: https://doi.org/10.1007/s00220-002-0775-3