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Ergodic isospectral theory of the Lax pairs of Euler equations with harmonic analysis flavor
Author(s):
Y.
Charles
Li
Journal:
Proc. Amer. Math. Soc.
133
(2005),
2681-2687.
MSC (2000):
Primary 35P05, 46N20, 76B99;
Secondary 37A30, 42A99
Posted:
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:
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- 2.
- Y. Li and A. Yurov, Lax pairs and Darboux transformations for Euler equations, Stud. Appl. Math. 111 (2003), 101. MR 1985997 (2004c:37191)
- 3.
- B. Koopman, Hamiltonian systems and transformations in Hilbert space, Proc. N. A. S. 17 (1931), 315.
- 4.
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- E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993, pp. 332, 334. MR 1232192 (95c:42002)
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- T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Lecture Notes in Math., Springer 448 (1975), 25. MR 0407477 (53:11252)
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- Y. Li, On 2D Euler equations. I. On the energy-Casimir stabilities and the spectra for linearized 2D Euler equations, J. Math. Phys. 41, no.2 (2000), 728. MR 1737017 (2001g:37129)
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- R. Shvidkoy, Y. Latushkin, The essential spectrum of the linearized 2D Euler operator is a vertical band, Contemp. Math. 327 (2003), 299. MR 1991549 (2004d:76025)
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Additional Information:
Y.
Charles
Li
Affiliation:
Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email:
cli@math.missouri.edu
DOI:
10.1090/S0002-9939-05-07828-7
PII:
S 0002-9939(05)07828-7
Keywords:
Lax pair,
Euler equation,
ergodic theorem,
oscillatory integral,
isospectral theory
Received by editor(s):
March 12, 2004
Received by editor(s) in revised form:
April 23, 2004
Posted:
March 22, 2005
Communicated by:
Carmen C. Chicone
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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