A Brief Biography of
George R. Sell
Victor A. Pliss
St Petersburg State University, Russia
George Roger Sell joined the Mathematics Faculty at
the University of
Minnesota in 1964, following two years as a Benjamin Peirce
Instructor at Harvard
At Minnesota, he was the co-founder, with Hans Weinberger,
of The Institute for
Mathematics and Its Applications (IMA) and served as the Associate
Director of the IMA for several years.
Sell ranks teaching and research as his most important
activities and has been the adviser to 17 Ph.D. candidates.
However, he has also directed the Computing Research Center,
has twice served on the University Senate, and has been a
Program Director at the National Science Foundation in
Washington, D.C. He founded, in 1989,
the Journal of Dynamics and Differential Equations and
is the editor of the Journal.
Born in 1937, in Milwaukee, Wisconsin, he is the oldest of the
eight children of Alice Sell and George P. Sell,
a machinist who thought Mathematics was important enough
to teach the children some Algebra while they were
still in Elementary School.
Sell was encouraged to pursue Mathematics by one of his
High School teachers,
Fr. Laurence McCall. He did so at Marquette University in
Milwaukee, graduating Summa Cum Laude in 1957, with majors
in Mathematics, Physics and Philosophy.
He received his M.Sc. in Mathematics in 1958, also from Marquette,
where his mentor was Professor Lester Heider, S. J.
He earned his Ph.D. in 1962 from the University of Michigan,
which he attended as the recipient of the General Electric Scholarship.
A student of Professor Wilfred Kaplan, Sell was awarded the prize
for the best dissertation written that year.
Since coming to Minnesota he has spent two sabbaticals at
The University of Southern
California, collaborating with Professor Robert Sacker, and
Dini in Florence, Italy, working with Professor Roberto Conti.
In addition to regular exchanges with Russian Universities,
Sell has had quarter leaves to the
Canberra, Australia, Kobe University, Japan, the
Polish Academy of Sciences, Warsaw,
Paris-Sud, Orsay, and the Universities of
Perugia and of
Palermo in Italy.
A sought-after conference speaker, Sell gave an invited address
at the 1982/83 International Congress of Mathematicians.
He has also given a plenary address at a joint AMS/MAA National
A conference in Sell's honor was held in Medina del Campo, Spain,
organized by the University
of Valladolid, in July, 2002. Twelve years earlier, Leningrad
State University (now St.
Petersburg State University) awarded him the degree, Honorary
Doctor of Science. Until then only four persons had been so
Sell and his wife, Gerry, a Marquette classmate, are the parents
of six children, the grandparents of five, and celebrated
their 45th wedding anniversary in June, 2003.
Many researchers in dynamical systems consider Sell to be
the Leader of the use of skew-product flows for the study of
the dynamics of solutions of nonautonomous differential equations.
In 1967 he published a seminal work  on this
topic. By using various topologies, he observed that under
reasonable assumptions, the space of all time-translates
of the coefficients of the equations has compact closure
in an appropriate function space.
In this way, the time-axis is imbedded into a compact space
with a flow on it. (This is a significant extension of an
earlier work of Miller, who used this approach for the
study of differential equations with almost periodic coefficients.)
At a later time, Sell and others began to refer to this
approach to nonautonomous equations as skew-product
|This photograph was reproduced from
the book "I Have a Photographic Memory" by Paul Halmos (published by
the AMS), where the
header states: F.P. Greenleaf and G.R. Sell. Another San Francisco
picture. Fred(left) is interested in invariant means on topological
groups and George in the periodicity properties of solutions of
ordinary differential equations. The photo was taken in the
One of the earliest applications of the skew-product structure
was in a joint paper with Markus  in which
the space-rescue problem by a satellite in a quasi-periodic
gravitational field of N bodies is shown to be solvable.
This problem reduces to a problem of controllability in a
time-dependent vector field, where the skew-product structure comes to
In the early 1970s, Sell began a major collaboration with
Sacker on several aspects of dynamics.
This collaboration began with a basic paper on
topological dynamics in which they seek a necessary and
sufficient condition that an extension of a minimal set
be a covering space for that minimal set. In particular,
they show that, if the action group is compactly generated,
then a necessary and sufficient condition for the
covering space is that the cardinality of the fiber of the
extension be the same finite number for each point in
the minimal set . At a later time, they use
theorem of topological dynamics to present a unified theory
on the existence of almost periodic solutions of differential
equations with almost periodic coefficients .
An earlier variant of this work appears in  in
which Sell shows that a weak asymptotic stability condition for a
bounded solution implies that this bounded solution is periodic.
Sell and Sacker have several very important publications in
the general area of linear skew-product flows over a compact
base. One of the issues they address in [6, 7] is
to determine necessary and sufficient conditions for an
exponential dichotomy. As is known, a necessary condition
for an exponential dichotomy is that the fiber of the
bounded space contains only the 0-vector over each base point.
The question then becomes: is this also a sufficient condition?
What they derive is an Alternative Theorem, which states
that: either there exists an exponential dichotomy over the
entire base space, or the flow on the base space has a
gradient-like structure. Furthermore, when the flow on the
base space is gradient-like, they find additional conditions,
which imply the existence of an exponential dichotomy.
Altogether, this work gives a definitive statement on
the theory of the existence of exponential dichotomies.
In 1994, Sell and Sacker extended this theory to
infinite-dimensional skew-product semiflows, where each
fiber is a Banach space and the solution operator in the
Banach space has an asymptotic compactness property
. This extension is applicable in the case of
parabolic and hyperbolic partial differential equations with
The concept of the Sacker-Sell spectrum grew out of their
joint work on the spectral theory for linear differential systems . The main aspect of this theory,
in the context of finite-dimensional ordinary differential equations,
is that the vector bundle for the linear skew-product
flow be decomposed into a finite sum of invariant subbundles
(a Whitney sum) and that each sub-bundle is associated with
a unique spectral interval. The Sacker-Sell spectrum,
which is defined as those values of the shift-parameter
for which the shifted linear skew-product flow does not
have an exponential dichotomy, is precisely the union of
these spectral intervals. This spectral theory has proven
to be a useful tool in the study of the dynamics of compact
invariant submanifolds for nonlinear skew-product flows,
see , as well as [11, 12].
The connection between the invariant sub-bundles, with the
Sacker-Sell spectrum, and the
Oseledec-Millionscikov Multiplicative Ergodic Theorem,
with the related Lyapunov exponents, is developed in a
joint paper with Sell, Johnson, and Palmer .
This paper also contains a geometric proof of the
Multiplicative Ergodic Theorem.
An interesting related development was in the effort of Sell
to extend the Hopf-Sacker bifurcation theory to the
bifurcation of higher-dimensional tori .
The Sacker-Sell spectral theory for invariant manifolds is
especially suited for this analysis, which in turn gives a
rigorous basis for examining the Hopf-Landau route to turbulence.
In , Sell shows that these higher-dimensional
bifurcations can occur near the Ruelle-Takens strange attractors.
The theory of Melnikov for proving the transversal
intersection of the stable and unstable manifolds,
under a small time-periodic perturbation of an autonomous
differential equation, and the subsequent analysis of
the chaotic behavior of the induced flow, are topics of
great interest in dynamics.
However, this method lacked a clear extension to
nonperiodic perturbations. In 1989,
Sell and Meyer showed that a complete extension of
the Melnikov theory to the case of an almost periodic
perturbation is possible . This theory is
built on the Sacker-Sell spectral theory and the related concepts of
Sell began to examine the issues of infinite-dimensional
dynamics in evolutionary equations in the 1980s.
His first contribution in this area is the joint work with
Foias and Temam on inertial manifolds .
They found a Gap Condition, which guarantees the existence of
inertial manifolds. Applications to reaction diffusion equations and
the Kuramoto-Sivashinsky equation were presented here and in
, with Nicolaenko joining them as a
Interest in the study of inertial manifolds grew quickly
in the late 1980s. Among other works,
Sell, with Fabes and Luskin, showed that the
Sacker approach of elliptic regularization can be extended
to the theory of inertial manifolds .
Also, in the case of the reaction diffusion equations,
the Gap Condition for inertial manifolds, which is
satisfied in low space-dimension, fails in three dimensions.
As a result, the paper  by Sell and
Mallet-Paret is a very important contribution to the theory of
inertial manifolds. Instead of the Gap Condition, they use
a concept called the Principle of Spatial Averaging for a
reaction diffusion equation on a suitable three-dimensional
domain to prove the existence of inertial manifolds.
While the Principle of Spatial Averaging is an alternative
to the Gap Condition in three dimensions, it fails in four dimensions.
In the paper , with Z Shao joining as a
co-author, they construct an example of a reaction-diffusion equation
in four dimensions that does not have a normally hyperbolic inertial
Another area in which Sell excelled is in the study of the
Navier-Stokes equations. He collaborated with Raugel
in an in-depth study of the Navier-Stokes equations on
thin three-dimensional domains, see [22, 23, 24].
This work is notable for two reasons: First, they show
the existence of globally regular solutions for data much
larger than that found in other then existing theories.
Second, they show the existence of a global attractor
for the weak (Leray) solutions of the Navier-Stokes equations
on a thin three-dimensional domain. Furthermore, this
global attractor consists entirely of globally regular
strong solutions. It seems that this may be the first known
theory of a three-dimensional problem where the global attractor
has this regularity property. As noted in the recent book by
Sell and You , this regularity is essential
for the use of long-time dynamics techniques in the study of
the Navier-Stokes equations.
This book is a major contribution to the literature on the
dynamics of infinite-dimensional problems.
It should benefit researchers for many years to come.
We mentioned above the global attractor for the weak solutions
of the Navier-Stokes equations. The existence of such
an attractor for any three-dimensional problem on a
smooth bounded domain is proved by Sell in .
By using a technique he originally developed for use in
the theory of ordinary differential equations
, Sell is able
to sidestep the two main obstacles for studying the
long-time dynamics of the three-dimensional Navier-Stokes equations:
(1) the possible lack of uniqueness of weak solutions, and (2)
the possible lack of regularity of the weak solutions.
In the last few years, Sell has again joined with
Mallet-Paret in publishing two works on the long-time dynamics
of solutions of systems of delay-differential equations
with a feedback property, [28, 29].
For this infinite-dimensional problem, they show: (1) a discrete
sign-change function is a Lyapunov function for the dynamics, and
(2) the Poincare-Bendixson Theorem is applicable in the sense that the
omega-limit set of every bounded solution is imbeddable in the
Another important paper of Sell is his work on the smooth
linearization of a vector field in the vicinity of a fixed
point . The objective in this paper is to
extend the Sternberg Theorem, which treats the \(C^\infty\)-linearization near a fixed point, to handle a finite
level of smoothness.
Finally I come to my collaborations with George.
These works include the publications [31, 32, 33,
34] plus two notes in Russian journals.
In this series of papers, we study the perturbation theory
of a class of invariant sets that include invariant manifolds
and Anosov flows. Furthermore, this class is closed under any
finite number of set products.
The term laminated invariant sets is a good way to describe
these sets. By imposing a weak form of
normal hyperbolicity on the unperturbed problem,
we show that, under a small perturbation of the
vector field in the vicinity of a laminated invariant set,
the perturbed equation has a homeomorphic laminated invariant set
nearby, and the perturbed set inherits the weak form of
normal hyperbolicity. In this way, the laminated
invariant sets are points of continuity for the dynamics.
This general theory of approximation dynamics is carried out
for finite-dimensional ordinary differential equations in
[31, 32]. The infinite-dimensional theory is
addressed in [33, 34], with applications to the
Navier-Stokes equations presented in .
I have known George and his family since 1967, when we
first met at the University of Southern California.
I remember the warm hospitality shown to me by George and Gerry.
One great event was the invitation to come to their home
for a "Minnesota feast", complete with turkey and stuffing.
I have cherished this friendship with George and Gerry ever since.
Our collaborations in research began in 1989/90 when
I was visiting the IMA at the University of Minnesota.
When I arrived, George described to me his concept of a
laminated invariant set and suggested that we collaborate
on the study of the dynamics near such a set.
It quickly became clear to us that this was to be a
major project. I have enjoyed working with George.
He has great mathematical insight and a marvelous sense
of humor, which enabled us to surmount some of the bumps.
I am grateful for the opportunity given to me for this
collaboration, since it has led me to do some of my
I am confident that this is an experience shared by many others.
||G. R. Sell, "Nonautonomous differential equations as
dynamical systems: I and II,"
Trans. Amer. Math. Soc. 127 (1967): 241-262 and
||L. Markus and G. R. Sell, "Control in conservative
dynamical systems. Recurrence and capture in aperiodic fields,"
J. Differential Equations 16 (1974): 472-505.
||R. J. Sacker and G. R. Sell, "Finite extensions of minimal
Trans. Amer. Math. Soc. 190 (1974): 429-458.
||R. J. Sacker and G. R. Sell, Lifting Properties in
Skew-Product Flows with Applications to Differential Equations,
Memoirs Amer. Math. Soc. 190 (1977).
||G. R. Sell, "Periodic solutions and asymptotic stability,"
J. Differential Equations 2 (1966): 143-157.
||R. J. Sacker and G. R. Sell, "Existence of dichotomies and
invariant splittings for linear differential systems I,"
J. Differential Equations 15 (1974): 429-458.
||R. J. Sacker and G. R. Sell, "Existence of dichotomies and
invariant splittings for linear differential systems II,"
J. Differential Equations 22 (1976): 478-496.
||R. J. Sacker and G. R. Sell, "Dichotomies in linear
evolutionary equations in Banach spaces,"
J. Differential Equations 113 (1994): 17-67.
||R. J. Sacker and G. R. Sell, "A spectral theory for
linear differential systems,"
J. Differential Equations 27 (1978): 320-358.
||R. J. Sacker and G. R. Sell, "The spectrum of an
J. Differential Equations 38 (1980): 135-160.
||G. R. Sell, "Hyperbolic almost periodic
solutions and toroidal limit sets,"
Proc. Nat. Acad. Sci. USA 74 (1977): 3124-3125.
||G. R. Sell, "The structure of a flow
in the vicinity of an almost periodic motion,"
J. Differential Equations 27 (1978): 359-393.
||R. A. Johnson, K. J. Palmer, and G. R. Sell,
"Ergodic properties of linear dynamical systems,"
SIAM J. Math. Anal. 18 (1987): 1-33.
||G. R. Sell, "Bifurcation of higher dimensional tori,"
Arch. Rational Mech. Anal. 69 (1979): 199-230.
||G. R. Sell, "Hopf-Landau bifurcation near strange attractors,"
in Chaos and Order in Nature, pp 84-91, Springer Verlag,
||K. R. Meyer and G. R. Sell, "Melnikov transformations,
Bernoulli bundles and almost periodic perturbations,"
Trans. Amer. Math. Soc. 314 (1989): 63-105.
||C. Foias, G. R. Sell, and R. Temam, "Inertial manifolds
for nonlinear evolutionary systems,"
J. Differential Equations 73 (1988): 309-353.
||C. Foias, B. Nicolaenko, G. R. Sell, and R. Temam,
"Inertial manifolds for the Kuramoto-Sivashinsky equation
and an estimate of their lowest dimension,"
J. Math. Pures Appl. 67 (1988): 197-226.
||E. Fabes, M. Luskin, and G. R. Sell, "Construction of
inertial manifolds by elliptic regularization,"
J. Differential Equations 89 (1991): 355-387.
||J. Mallet-Paret and G. R. Sell, "Inertial manifolds
for reaction-diffusion equations in higher space dimensions,"
J. Amer. Math. Soc. 1 (1988): 805-866.
||J. Mallet-Paret, G. R. Sell, and Z. Shao,
"Obstructions for the existence of normally hyperbolic
Indiana J. Math. 42 (1993): 1027-1055.
||G. Raugel and G. R. Sell, "Navier-Stokes equations on
thin 3D domains I: Global attractors and global regularity of
J. Amer. Math. Soc. 6 (1993): 503-568.
||G. Raugel and G. R. Sell, "Navier-Stokes equations on
thin 3D domains II: Global regularity of spatially periodic
in H. Brezis and J. L. Lions (Eds) Nonlinear Partial Differential
Equations and Their Applications, College de France Seminar, Vol
11, Pitman Research Notes in Mathematics Series, No 299, pp.
205-247, Longman (1994).
||G. Raugel and G. R. Sell, "Navier-Stokes equations on
thin 3D domains III: Global and local attractors,"
in Turbulence in Fluid Flows: A Dynamical Systems Approach,
IMA Volumes in Mathematics and its Applications, Vol 55, pp.
137-163, Springer Verlag (1993).
||G. R. Sell and Y. You, Dynamics of Evolutionary
Equations, Applied Math. Sciences, Vol 143, Springer
||G. R. Sell, "Global attractors for the 3-dimensional
J. Dynamics and Differential Equations 8 (1996):
||G. R. Sell, "Differential equations without uniqueness
and classical topological dynamics,"
J. Differential Equations 14 (1973): 42-56.
||J. Mallet-Paret and G. R. Sell, "Systems of delay
differential equations I: Floquet multipliers and discrete Lyapunov
J. Differential Equations 125 (1996): 385-440.
||J. Mallet-Paret and G. R. Sell, "The Poincare-Bendixson
theorem for monotone cyclic feedback systems with delay,"
J. Differential Equations 125 (1996): 441-489.
||G. R. Sell, "Smooth linearization near a fixed point,"
Amer. J. Math. 107 (1985): 1035-1091.
||V. A. Pliss and G. R. Sell, "Perturbations of attractors
of differential equations,"
J. Differential Equations 92 (1991): 100-124.
||V. A. Pliss and G. R. Sell, "Approximation dynamics and the
stability of invariant sets,"
J. Differential Equations 149 (1998): 1-52.
||V. A. Pliss and G. R. Sell, "Robustness of exponential
dichotomies in infinite dimensional dynamical systems,"
J. Dynamics Differential Equations 11 (1999):
||V. A. Pliss and G. R. Sell, "Perturbations of normally
hyperbolic manifolds with applications to the Navier-Stokes
J. Differential Equations 169 (2001):