Approximation of Stochastic Invariant Manifolds: Stochastic Manifolds for Nonlinear SPDEs I

136 Pages·2015·3.24 MB·English

by  Mickaël D. Chekroun

SPRINGER BRIEFS IN MATHEMATICS
Mickaël D. Chekroun
Honghu Liu
Shouhong Wang
Approximation of
Stochastic Invariant
Manifolds
Stochastic Manifolds
for Nonlinear SPDEs I
SpringerBriefs in Mathematics
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Micka l D. Chekroun Honghu Liu
(cid:129)
Shouhong Wang
Approximation of Stochastic
Invariant Manifolds
Stochastic Manifolds for Nonlinear SPDEs I
123
Mickaël D.Chekroun ShouhongWang
Universityof California Indiana University
Los Angeles, CA Bloomington,IN
USA USA
Honghu Liu
Universityof California
Los Angeles, CA
USA
ISSN 2191-8198 ISSN 2191-8201 (electronic)
SpringerBriefs inMathematics
ISBN 978-3-319-12495-7 ISBN 978-3-319-12496-4 (eBook)
DOI 10.1007/978-3-319-12496-4
LibraryofCongressControlNumber:2014956373
Mathematics Subject Classification: 37L65, 37D10, 37L25, 35B42, 37L10, 37L55, 60H15, 35R60,
34F05,34G20,37L05
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To our families
AnapproximationofastochastichyperbolicmanifoldassociatedwithastochasticBurgers-type
equation:x-disintegrationovertheleadingpairofstable/unstablemodes,foragivenrealization
.
ofthenoise.Thecalculationsarebasedontheformulasderivedinthismonograph;seeChap.7
Preface
Theintriguingfigurefromthepreviouspagepresentsavisualizationofthetypeof
manifolds this monograph is concerned with. First, obtained as a graph over some
chosen linearized spatial modes from a nonlinear stochastic partial differential
equation (SPDE), such a manifold—and its geometric properties—depends natu-
rally on the spatial variable. Second, the shape of this manifold is nonlinear as a
fingerprint of the nonlinear effects conveyed by the nonlinear SPDE. Third, this
manifoldisstochastic,inotherwords,itdependsontimeforagivenrealizationof
the noise, meaning that “mutations” of the shape—such as changes in the
(Gaussian) curvature—can occur as time flows. Manifolds that share these three
features and more importantly for which the dynamics of the underlying nonlinear
SPDEiseitherattractedtoormeandersaround,arethemanifoldsofinterestinthis
two-volume series.
In this respect, a pathwise approach from the theory of random dynamical
systems (RDS)isadoptedinboth volumes.Volume Ideals with approximation of
stochastic manifolds that are invariant for the SPDE dynamics, while Volume II
[37] deals with stochastic manifolds that may not be invariant nor attractive and
that can still capture essential features of the dynamics through appropriate
parameterizations of the small spatial scales by the large ones. The small-scale
parameterizations proposed in Volume II are articulated around a new concept of
stochastic manifolds, namely the stochastic parameterizing manifolds (PMs).
An important common feature is shared by the (pathwise) approximation
formulas derived in Volume I and the (pathwise) parameterization techniques
introducedinVolumeII:botharecharacterizedaspullbacklimitsfrombackward–
forward systems which are only partially coupled, facilitating the calculation of
such limits, either analytically or numerically.
The aforementioned pullback limits arise under the form of Lyapunov–Perron
integrals which are useful for the rigorous treatment of the problem of approxi-
mation to the leading order,1 of important stochastic manifolds such as—but not
1 WithrespecttothenonlinearityinvolvedintheSPDEathand.
ix
x Preface
limited to—stochastic critical manifolds2 built as random graphs over a fixed
number of critical modes which lose their stability as a control parameter varies.
In this respect, Chaps. 6 and 7 contain explicit formulas for the leading-order
Taylor approximation of stochastic (local) critical manifolds or more general sto-
chastic hyperbolic manifolds. The pullback characterization of these formulas
provides a useful interpretation of the corresponding approximating manifolds
which gives rise to a simple framework that allows us, furthermore, to unify the
previous approximation approaches of stochastic invariant manifolds, as discussed
in Volume II; see [37, Sect. 4.1].
To help the reader appreciate this unification as well as the more prospective
material presented in Volume II, we took the freedom to include a self-contained
(short)surveyonthetheoryofexistenceandattractionofone-parameterfamiliesof
stochastic invariant manifolds in Chaps. 4 and 5.
Los Angeles, September 2014 Mickaël D. Chekroun
Los Angeles Honghu Liu
Bloomington Shouhong Wang
2 Includingstochasticmanifoldssuchasthecenterorcenter-unstablemanifolds.
Acknowledgments
Preliminary versions of this work were presented by the authors at the AIMS
SpecialSessionon“AdvancesinClassicalandGeophysicalFluidDynamics,”held
at the Ninth AIMS Conference on Dynamical Systems, Differential Equations and
Applications in Orlando, July 2012; at the “workshop on Random Dynamical
Systems,”heldattheInstituteofMathematicsandApplicationsinOctober2012;at
the “Lunch Seminars” held at the Center for Computational and Applied Mathe-
matics, Purdue University, in November 2012; at the “Applied Mathematics
Seminar,” held at the University of Illinois at Chicago in March 2013; at AMS
Special Session on “Partial Differential Equations from Fluid Mechanics,” held at
University of Louisville in October 2013; at the “PDE seminar,” held at Indiana
University in October 2013; and at the Alpine Summer School on Dynamics,
Stochastics, and Predictability of the Climate System, Valle d’Aosta, Italy in June
2014.
We would like to thank all those who helped in the realization of this book
through encouragement, advice, or scientific exchanges: Jerry Bona, Jerome
Darbon,JinqiaoDuan,MichaelGhil,NathanGlatt-Holtz,DmitriKondrashov,Tian
Ma,JamesMcWilliams,DavidNeelin,JamesRobinson,JeanRoux,EricSimonnet,
Roger Temam, Yohann Tendero, and Kevin Zumbrun. More particularly, the
authors are grateful to Michael Ghil for his constant support and interest in this
work, overthe years. MDC isgrateful toJames McWilliams andDavid Neelin for
stimulating discussions on the closure problem of turbulence and stochastic
parameterizations; and to Michael Ghil and Roger Temam for stimulating discus-
sionsontheslowandthe“fuzzy”manifold.MDCthanksalsoJinqiaoDuanforthe
reference [103], and for discussions on stochastic invariant manifolds.
We would like to express also our thanks to Ute McCrory (Springer) for her
invaluable patience and support during the preparation of this book.
MDC and HL are supported by the National Science Foundation grant DMS-
1049253 and Office of Naval Research grant N00014-12-1-0911. SW is supported
in part by National Science Foundation grants DMS-1211218 and DMS-1049114,
and by Office of Naval Research grant N00014-11-1-0404. SW is most grateful to
xi