## Ergodic isospectral theory of the Lax pairs of Euler equations with harmonic analysis flavor

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- by Y. Charles Li
- Proc. Amer. Math. Soc.
**133**(2005), 2681-2687 - DOI: https://doi.org/10.1090/S0002-9939-05-07828-7
- Published electronically: March 22, 2005
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## Abstract:

Isospectral theory of the Lax pairs of both 3D and 2D Euler equations of inviscid fluids is developed. Eigenfunctions are represented through an ergodic integral. The Koopman group and mean ergodic theorem are utilized. Further harmonic analysis results on the ergodic integral are introduced. The ergodic integral is a limit of the oscillatory integral of the first kind.## References

- Yanguang Li,
*A Lax pair for the two dimensional Euler equation*, J. Math. Phys.**42**(2001), no. 8, 3552–3553. MR**1845205**, DOI 10.1063/1.1378305 - Yanguang Li and Artyom V. Yurov,
*Lax pairs and Darboux transformations for Euler equations*, Stud. Appl. Math.**111**(2003), no. 1, 101–113. MR**1985997**, DOI 10.1111/1467-9590.t01-1-00229 - B. Koopman, Hamiltonian systems and transformations in Hilbert space,
*Proc. N. A. S.***17**(1931), 315. - Peter D. Lax,
*Functional analysis*, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. MR**1892228** - Elias M. Stein,
*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192** - Tosio Kato,
*Quasi-linear equations of evolution, with applications to partial differential equations*, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, pp. 25–70. MR**0407477** - Yanguang Li,
*On 2D Euler equations. I. On the energy-Casimir stabilities and the spectra for linearized 2D Euler equations*, J. Math. Phys.**41**(2000), no. 2, 728–758. MR**1737017**, DOI 10.1063/1.533176 - Roman Shvidkoy and Yuri Latushkin,
*The essential spectrum of the linearized 2D Euler operator is a vertical band*, Advances in differential equations and mathematical physics (Birmingham, AL, 2002) Contemp. Math., vol. 327, Amer. Math. Soc., Providence, RI, 2003, pp. 299–304. MR**1991549**, DOI 10.1090/conm/327/05822

## Bibliographic Information

**Y. Charles Li**- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 270213
- Email: cli@math.missouri.edu
- Received by editor(s): March 12, 2004
- Received by editor(s) in revised form: April 23, 2004
- Published electronically: March 22, 2005
- Communicated by: Carmen C. Chicone
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**133**(2005), 2681-2687 - MSC (2000): Primary 35P05, 46N20, 76B99; Secondary 37A30, 42A99
- DOI: https://doi.org/10.1090/S0002-9939-05-07828-7
- MathSciNet review: 2146214