Exponential averaging for Hamiltonian evolution equations

Authors:
Karsten Matthies and Arnd Scheel

Journal:
Trans. Amer. Math. Soc. **355** (2003), 747-773

MSC (2000):
Primary 37K55, 37L10, 70K65; Secondary 35Q55, 70K70

Published electronically:
October 2, 2002

MathSciNet review:
1932724

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrödinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.

**[Bam99]**Dario Bambusi,*Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations*, Math. Z.**230**(1999), no. 2, 345–387. MR**1676714**, 10.1007/PL00004696**[FeTi98]**Andrew B. Ferrari and Edriss S. Titi,*Gevrey regularity for nonlinear analytic parabolic equations*, Comm. Partial Differential Equations**23**(1998), no. 1-2, 1–16. MR**1608488**, 10.1080/03605309808821336**[FiVi00]**B. Fiedler and M. I. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms, MSAP preprint (2000).**[FoTe89]**C. Foias and R. Temam,*Gevrey class regularity for the solutions of the Navier-Stokes equations*, J. Funct. Anal.**87**(1989), no. 2, 359–369. MR**1026858**, 10.1016/0022-1236(89)90015-3**[LoMe88]**P. Lochak and C. Meunier,*Multiphase Averaging for Classical Systems*, Applied Mathematical Sciences**72**, Springer-Verlag, New York (1988).**[MaRa94]**Jerrold E. Marsden and Tudor S. Ratiu,*Introduction to mechanics and symmetry*, 2nd ed., Texts in Applied Mathematics, vol. 17, Springer-Verlag, New York, 1999. A basic exposition of classical mechanical systems. MR**1723696****[Mat00]**K. Matthies, Homogenization of exponential order for elliptic systems in infinite cylinders, preprint (2000).**[Mat01]**K. Matthies, Time-averaging under fast periodic forcing of parabolic partial differential equations: Exponential estimates,*J. Differential Equations***174**(2001), 133-180.**[Nei84]**A. I. Neĭshtadt,*The separation of motions in systems with rapidly rotating phase*, Prikl. Mat. Mekh.**48**(1984), no. 2, 197–204 (Russian); English transl., J. Appl. Math. Mech.**48**(1984), no. 2, 133–139 (1985). MR**802878**, 10.1016/0021-8928(84)90078-9**[Nek79]**N. N. Nekhorosev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, (Russian)*Uspehi Mat. Nauk***32**(1977), no. 6 (198), 5-66.**[Paz83]**A. Pazy,*Semigroups of linear operators and applications to partial differential equations*, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR**710486****[Poe99]**Jürgen Pöschel,*On Nekhoroshev estimates for a nonlinear Schrödinger equation and a theorem by Bambusi*, Nonlinearity**12**(1999), no. 6, 1587–1600. MR**1726666**, 10.1088/0951-7715/12/6/310**[Pro91]**Keith Promislow,*Time analyticity and Gevrey regularity for solutions of a class of dissipative partial differential equations*, Nonlinear Anal.**16**(1991), no. 11, 959–980. MR**1106997**, 10.1016/0362-546X(91)90100-F**[SaScWu99]**B. Sandstede, A. Scheel, and C. Wulff,*Bifurcations and dynamics of spiral waves*, J. Nonlinear Sci.**9**(1999), no. 4, 439–478. MR**1700672**, 10.1007/s003329900076**[TBDHT96]**P. Takáč, P. Bollerman, A. Doelman, A. van Harten, and E. S. Titi,*Analyticity of essentially bounded solutions to semilinear parabolic systems and validity of the Ginzburg-Landau equation*, SIAM J. Math. Anal.**27**(1996), no. 2, 424–448. MR**1377482**, 10.1137/S0036141094262518**[Vis00]**M. Vishik, Attractors of evolution equations with rapidly oscillating terms, Proc. Internat. Conf. in Honor of R. Temam, Paris, to appear.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
37K55,
37L10,
70K65,
35Q55,
70K70

Retrieve articles in all journals with MSC (2000): 37K55, 37L10, 70K65, 35Q55, 70K70

Additional Information

**Karsten Matthies**

Affiliation:
Mathematical Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Address at time of publication:
FU Berlin, Institut für Mathematik I, Arnimallee 2-6, 14195 Berlin, Germany

Email:
matthies@maths.warwick.ac.uk, matthies@math.fu-berlin.de

**Arnd Scheel**

Affiliation:
School of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, Minnesota 55455

Email:
scheel@math.umn.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03143-4

Keywords:
Averaging,
exponential order,
analytic evolution equation,
infinite-dimensional Hamiltonian system,
Gevrey regularity,
nonlinear Schr\"odinger equation

Received by editor(s):
October 8, 2001

Received by editor(s) in revised form:
May 15, 2002

Published electronically:
October 2, 2002

Additional Notes:
The first author was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant Ma 2351/1

The second author gratefully acknowledges support from DAAD/Procope, Nr. D/0031082

Article copyright:
© Copyright 2002
American Mathematical Society