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 Mat. Sb., 2001, Volume 192, Number 3, Pages 137–160 (Mi msb554)

This article is cited in 3 scientific papers (total in 3 papers)

Periodic differential equations with self-adjoint monodromy operator

V. I. Yudovich

Rostov State University

Abstract: A linear differential equation $\dot u=A(t)u$ with $p$-periodic (generally speaking, unbounded) operator coefficient in a Euclidean or a Hilbert space $\mathbb H$ is considered. It is proved under natural constraints that the monodromy operator $U_p$ is self-adjoint and strictly positive if $A^*(-t)=A(t)$ for all $t\in\mathbb R$.
It is shown that Hamiltonian systems in the class under consideration are usually unstable and, if they are stable, then the operator $U_p$ reduces to the identity and all solutions are $p$-periodic.
For higher frequencies averaged equations are derived. Remarkably, high-frequency modulation may double the number of critical values.
General results are applied to rotational flows with cylindrical components of the velocity $a_r=a_z=0$, $a_\theta=\lambda c(t)r^\beta$, $\beta<-1$,   $c(t)$ is an even $p$-periodic function, and also to several problems of free gravitational convection of fluids in periodic fields.

DOI: https://doi.org/10.4213/sm554

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English version:
Sbornik: Mathematics, 2001, 192:3, 455–478

Bibliographic databases:

UDC: 517.98
MSC: 34G10, 34A30, 76D05, 76E06
Received: 14.11.1999 and 24.08.2000

Citation: V. I. Yudovich, “Periodic differential equations with self-adjoint monodromy operator”, Mat. Sb., 192:3 (2001), 137–160; Sb. Math., 192:3 (2001), 455–478

Citation in format AMSBIB
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This publication is cited in the following articles:
1. V. I. Yudovich, “Eleven great problems of mathematical hydrodynamics”, Mosc. Math. J., 3:2 (2003), 711–737
2. V. B. Levenshtam, “Justification of the averaging method for parabolic equations containing rapidly oscillating terms with large amplitudes”, Izv. Math., 70:2 (2006), 233–263
3. A. K. Kapikyan, V. B. Levenshtam, “First-order partial differential equations with large high-frequency terms”, Comput. Math. Math. Phys., 48:11 (2008), 2059–2076
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