Ergodic isospectral theory of the Lax pairs of Euler equations with harmonic analysis flavor
Author:
Y. Charles Li
Journal:
Proc. Amer. Math. Soc. 133 (2005), 26812687
MSC (2000):
Primary 35P05, 46N20, 76B99; Secondary 37A30, 42A99
Published electronically:
March 22, 2005
MathSciNet review:
2146214
Fulltext PDF Free Access
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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.
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Yanguang
Li and Artyom
V. Yurov, Lax pairs and Darboux transformations for Euler
equations, Stud. Appl. Math. 111 (2003), no. 1,
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(2004c:37191), http://dx.doi.org/10.1111/14679590.t01100229
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(2003a:47001)
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(95c:42002)
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 1.
 Y. Li, A Lax pair for the two dimensional Euler equation, J. Math. Phys. 42, no.8 (2001), 3552. MR 1845205 (2002f:35188)
 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.
 P. Lax, Functional Analysis, John Wiley & Sons, Inc., 2002, pp. 445, 443, 440. MR 1892228 (2003a:47001)
 5.
 E. Stein, Harmonic Analysis: RealVariable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993, pp. 332, 334. MR 1232192 (95c:42002)
 6.
 T. Kato, Quasilinear equations of evolution, with applications to partial differential equations, Lecture Notes in Math., Springer 448 (1975), 25. MR 0407477 (53:11252)
 7.
 Y. Li, On 2D Euler equations. I. On the energyCasimir stabilities and the spectra for linearized 2D Euler equations, J. Math. Phys. 41, no.2 (2000), 728. MR 1737017 (2001g:37129)
 8.
 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:
http://dx.doi.org/10.1090/S0002993905078287
PII:
S 00029939(05)078287
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
Published electronically:
March 22, 2005
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
