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), 2681-2687
MSC (2000): Primary 35P05, 46N20, 76B99; Secondary 37A30, 42A99
Published electronically: March 22, 2005
MathSciNet review: 2146214
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.
- 1. Yanguang Li, A Lax pair for the two dimensional Euler equation, J. Math. Phys. 42 (2001), no. 8, 3552–3553. MR 1845205, https://doi.org/10.1063/1.1378305
- 2. 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, https://doi.org/10.1111/1467-9590.t01-1-00229
- 3. B. Koopman, Hamiltonian systems and transformations in Hilbert space, Proc. N. A. S. 17 (1931), 315.
- 4. Peter D. Lax, Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2002. MR 1892228
- 5. 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
- 6. 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), Springer, Berlin, 1975, pp. 25–70. Lecture Notes in Math., Vol. 448. MR 0407477
- 7. 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, https://doi.org/10.1063/1.533176
- 8. 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, https://doi.org/10.1090/conm/327/05822
- Y. Li, A Lax pair for the two dimensional Euler equation, J. Math. Phys. 42, no.8 (2001), 3552. MR 1845205 (2002f:35188)
- Y. Li and A. Yurov, Lax pairs and Darboux transformations for Euler equations, Stud. Appl. Math. 111 (2003), 101. MR 1985997 (2004c:37191)
- B. Koopman, Hamiltonian systems and transformations in Hilbert space, Proc. N. A. S. 17 (1931), 315.
- P. Lax, Functional Analysis, John Wiley & Sons, Inc., 2002, pp. 445, 443, 440. MR 1892228 (2003a:47001)
- E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993, pp. 332, 334. MR 1232192 (95c:42002)
- 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)
- 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)
- 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)
Y. Charles Li
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
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.