# A Brief Biography of George R. Sell

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 A Brief Biography of George R. Sell Victor A. Pliss St Petersburg State University, Russia George Sell

George Roger Sell joined the Mathematics Faculty at the University of Minnesota in 1964, following two years as a Benjamin Peirce Instructor at Harvard University.

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 another at Instituto Ulisse Dini in Florence, Italy, working with Professor Roberto Conti.

In addition to regular exchanges with Russian Universities, Sell has had quarter leaves to the University of Canberra, Australia, Kobe University, Japan, the Polish Academy of Sciences, Warsaw, L'Université de 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 Meeting.

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 honored.

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 [1] 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 dynamics.

 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 1960s.

One of the earliest applications of the skew-product structure was in a joint paper with Markus [2] 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 the fore.

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 [3]. At a later time, they use this general theorem of topological dynamics to present a unified theory on the existence of almost periodic solutions of differential equations with almost periodic coefficients [4]. An earlier variant of this work appears in [5] 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 [8]. This extension is applicable in the case of parabolic and hyperbolic partial differential equations with time-varying coefficients.

The concept of the Sacker-Sell spectrum grew out of their joint work on the spectral theory for linear differential systems [9]. 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 [10], 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 [13]. 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 [14]. 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 [15], 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 [16]. This theory is built on the Sacker-Sell spectral theory and the related concepts of normal hyperbolicity.

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 [17]. 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 [18], with Nicolaenko joining them as a co-author.

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 [19]. 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 [20] 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 [21], 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 manifold.

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 [25], 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 [26]. By using a technique he originally developed for use in the theory of ordinary differential equations [27], 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 two-dimensional plane.

Another important paper of Sell is his work on the smooth linearization of a vector field in the vicinity of a fixed point [30]. 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 [34].

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 best mathematics. I am confident that this is an experience shared by many others.

References

 [1] G. R. Sell, "Nonautonomous differential equations as dynamical systems: I and II," Trans. Amer. Math. Soc. 127 (1967): 241-262 and 263-283. [2] L. Markus and G. R. Sell, "Control in conservative dynamical systems. Recurrence and capture in aperiodic fields," J. Differential Equations 16 (1974): 472-505. [3] R. J. Sacker and G. R. Sell, "Finite extensions of minimal transformation groups," Trans. Amer. Math. Soc. 190 (1974): 429-458. [4] R. J. Sacker and G. R. Sell, Lifting Properties in Skew-Product Flows with Applications to Differential Equations, Memoirs Amer. Math. Soc. 190 (1977). [5] G. R. Sell, "Periodic solutions and asymptotic stability," J. Differential Equations 2 (1966): 143-157. [6] 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. [7] 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. [8] R. J. Sacker and G. R. Sell, "Dichotomies in linear evolutionary equations in Banach spaces," J. Differential Equations 113 (1994): 17-67. [9] R. J. Sacker and G. R. Sell, "A spectral theory for linear differential systems," J. Differential Equations 27 (1978): 320-358. [10] R. J. Sacker and G. R. Sell, "The spectrum of an invariant submanifold," J. Differential Equations 38 (1980): 135-160. [11] G. R. Sell, "Hyperbolic almost periodic solutions and toroidal limit sets," Proc. Nat. Acad. Sci. USA 74 (1977): 3124-3125. [12] G. R. Sell, "The structure of a flow in the vicinity of an almost periodic motion," J. Differential Equations 27 (1978): 359-393. [13] R. A. Johnson, K. J. Palmer, and G. R. Sell, "Ergodic properties of linear dynamical systems," SIAM J. Math. Anal. 18 (1987): 1-33. [14] G. R. Sell, "Bifurcation of higher dimensional tori," Arch. Rational Mech. Anal. 69 (1979): 199-230. [15] G. R. Sell, "Hopf-Landau bifurcation near strange attractors," in Chaos and Order in Nature, pp 84-91, Springer Verlag, (1981). [16] K. R. Meyer and G. R. Sell, "Melnikov transformations, Bernoulli bundles and almost periodic perturbations," Trans. Amer. Math. Soc. 314 (1989): 63-105. [17] C. Foias, G. R. Sell, and R. Temam, "Inertial manifolds for nonlinear evolutionary systems," J. Differential Equations 73 (1988): 309-353. [18] 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. [19] E. Fabes, M. Luskin, and G. R. Sell, "Construction of inertial manifolds by elliptic regularization," J. Differential Equations 89 (1991): 355-387. [20] 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. [21] J. Mallet-Paret, G. R. Sell, and Z. Shao, "Obstructions for the existence of normally hyperbolic inertial manifolds," Indiana J. Math. 42 (1993): 1027-1055. [22] G. Raugel and G. R. Sell, "Navier-Stokes equations on thin 3D domains I: Global attractors and global regularity of solutions," J. Amer. Math. Soc. 6 (1993): 503-568. [23] G. Raugel and G. R. Sell, "Navier-Stokes equations on thin 3D domains II: Global regularity of spatially periodic solutions," 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). [24] 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). [25] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Math. Sciences, Vol 143, Springer Verlag (2002). [26] G. R. Sell, "Global attractors for the 3-dimensional Navier-Stokes equations," J. Dynamics and Differential Equations 8 (1996): 1-33. [27] G. R. Sell, "Differential equations without uniqueness and classical topological dynamics," J. Differential Equations 14 (1973): 42-56. [28] J. Mallet-Paret and G. R. Sell, "Systems of delay differential equations I: Floquet multipliers and discrete Lyapunov functions," J. Differential Equations 125 (1996): 385-440. [19] 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. [30] G. R. Sell, "Smooth linearization near a fixed point," Amer. J. Math. 107 (1985): 1035-1091. [31] V. A. Pliss and G. R. Sell, "Perturbations of attractors of differential equations," J. Differential Equations 92 (1991): 100-124. [32] V. A. Pliss and G. R. Sell, "Approximation dynamics and the stability of invariant sets," J. Differential Equations 149 (1998): 1-52. [33] V. A. Pliss and G. R. Sell, "Robustness of exponential dichotomies in infinite dimensional dynamical systems," J. Dynamics Differential Equations 11 (1999): 471-513. [34] V. A. Pliss and G. R. Sell, "Perturbations of normally hyperbolic manifolds with applications to the Navier-Stokes equations," J. Differential Equations 169 (2001): 396-492.
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