# Lions, Jacques-Louis

# LIONS, JACQUES-LOUIS

(*b*. Grasse, France, 2 May 1928; *d*. Paris, France, 17 May 2001),

*mathematics and applications of mathematics, analysis, applied analysis, scientific computing, control theory*.

Lions was a scientist of remarkable prescience and immense energy. His vision extended to the development of entire areas of mathematical science. He understood that mathematics could make a great contribution to science and he worked to see this realized. While Lions’s name is rarely attached to the specific results of his researches, his mathematical legacy is extremely important. He authored or co-authored twenty books and nearly 600 articles, and had a considerable influence on the development of the French school of applied mathematics, and on many mathematicians and many mathematical institutions worldwide. He was also active in other areas of scientific research, in industry, and in politics.

Jacques-Louis Lions was the only son of Honoré Lions and Annette Muller. For more than thirty years, his father was the mayor of the old and charming town of Grasse located in the hills above the Mediterranean seashore, in a region producing many of the flowers used by the French perfume industry. Lions grew up in Grasse and he was very attached to the town and to this part of France. In 1950 he married his lifelong companion, Andrée Olivier, whom he had met during World War II in the Resistance. Their only son, Pierre-Louis, was born in Grasse in 1956. Pierre-Louis, like his father, became a famous mathematician, and he was awarded one of the Fields Medals at the International Congress of Mathematicians in Zurich in 1994.

After high school Lions studied at a small one-year college that would become the Université de Nice. His potential was detected by an examiner who advised him to prepare for the competitive École Normale Supérieure in Paris, which Lions entered in 1947 and attended until 1950. After the Ecole Normale Supérieure, instead of becoming a schoolteacher (as did many of his fellow students), Lions chose to become a university teacher. He obtained a fellowship from the Centre national de la recherche scientifique (CNRS) and started research under the direction of Laurent Schwartz at the Université de Nancy. Also in 1950, Schwartz was awarded the Fields Medal at the International Congress of Mathematicians at Harvard University in Cambridge, Massachusetts, for his work on the theory of distributions. Schwartz thought that the theory of partial differential equations should be completely recast in the context of distribution theory. Lions was one of several students whom Schwartz directed to take this new approach, and Lions’s doctoral thesis developed what has become the standard variational theory of linear elliptic and evolution equations. From then on Lions developed his own research, working hard and without any interruption to the end, even when he had other important and time-consuming responsibilities.

Lions received a doctor of science degree in 1954 and was appointed maître de conférences (associate professor) at the Faculté des sciences of Nancy. Later he became professor and, in 1963, he was appointed professor at the Faculté des sciences at the University of Paris. In 1973 he became Professor at the famous Collège de France in Paris, holding the chair that was named, by his choice, “Analyse mathématique des systèmes et de leur contrôle” (“Mathematical analysis of systems and their control”). He retired from the Collège de France in 1998.

The mathematical work of Lions is at the same time very diverse and well unified. Part but not all of his work is contained in the three-volume collected works published in 2003. This article will describe his work following the titles of these three volumes.

**Partial Differential Equations (PDEs) and Interpolation** . This work started in the early 1950s; continuing his thesis work, it contains many of the building blocks of what would be Lions’s “Analyse des Systèmes.” He first addressed, alone or in collaboration, many issues in linear PDE theory and in distribution theory. In the late 1950s Lions started his first three major and lasting works. Under the influence of Jean Leray, Lions became interested in nonlinear partial differential equations, in particular the incompressible Navier-Stokes equations. The mathematical analysis of the Navier-Stokes equations, which had been inactive since the pioneering work of Leray in the 1930s, came back to life in 1951, when Eber-hard Hopf established the long-time existence of weak solutions for bounded domains in three dimensions. The contributions of Lions to the subject are twofold. He and Giovanni Prodi proved independently the uniqueness of weak solutions in two space dimensions, publishing the result together in 1959. Also, as part of his better understanding of evolution equations, Lions was able to shorten and make considerably more accessible the results of Hopf; he thus contributed, with Olga Aleksandrovna Ladyzhenskaya, James Serrin and others, to the beginning of the modern theory of mathematical fluid dynamics.

Also around this time, Lions began his work with Enrico Magenes on nonhomogeneous boundary value problems, which led eventually to the publication of a three-volume book in 1968. A wealth of results were developed as part of this work that necessitated a better understanding and many different characterizations of the Sobolev spaces (introduced by Sergei L. Sobolev in the 1930s), the systematic study of Sobolev spaces with fractional exponents, and the theory of linear elliptic and parabolic equations in such spaces.

The third major work started in that period—parallel to, and necessary for Lions’s two other projects—was the theory of interpolation between Hilbert or Banach spaces; that is, constructing a space intermediate in the sense of topology and set inclusion between two given spaces. Lions made substantial contributions to interpolation and he initiated the interpolation between Hilbert spaces in 1953. Although “linear,” the last two areas of research eventually proved to be very important for nonlinear problems.

Following an idea of George Minty and Felix Brow-der, Lions was in his subsequent work involved in the development of the theory of strongly nonlinear equations that are monotone in their highest arguments. In his only work with Jean Leray, he published in 1965 one of the most general results in that direction, extending and simplifying an earlier result of Mark Vishik. Also, together with Guido Stampacchia, he published in 1965 and 1967 two articles that laid the basis of the theory of variational inequalities. Subsequently Lions continued to develop this theory alone, and with Haim Brezis, and, later, with Georges Duvaut in a book devoted to the application of variational inequalities to many concrete and specific problems in continuum mechanics and physics (1972). Lions’s theoretical work on nonlinear PDEs is included in a book published in 1969 that was very exhaustive at the time.

**Numerical Analysis, Scientific Computing, and Applications** . Lions became interested in this subject in the early 1960s. At that point he was influenced by another of his intellectual mentors, John von Neumann, who had designed the first computers and started to use numerical methods for the solution of partial differential equations from fluid mechanics and meteorology. Lions dreamed, almost alone in France, that there was an important future for mathematics in that direction, and he threw himself into this subject. At that time the French mathematical school was almost exclusively engaged in the development of the Bourbaki program in pure mathematics, and it is one of Lions’s major achievements and vision to have predicted this development and to have made applied mathematics accepted in the French mathematical community. He did not publish much in the area at first, but he started the French school of numerical analysis. He taught numerical analysis at the Institute Blaise Pascal in Paris and at Ecole Polytechnique. His students of that time and their academic descendants represent a very significant part of the French numerical analysis community.

In his courses, Lions used from the beginning the variational theory of boundary value problems that he developed himself in his thesis and subsequently. This point of view was further developed in the first theses that he directed in numerical analysis, thus producing the appropriate framework for the development of the finite element methods and for many other important subsequent and contemporary developments in numerical analysis. This includes the thesis and work of Jean Céa on the use of variational methods in numerical analysis, and the works of Philippe Ciarlet and Pierre-Arnaud Raviart on the numerical analysis of the finite elements methods. In this way Lions also played an important indirect role in the research on the numerical analysis of PDEs. Lions himself subsequently published in the area of numerical analysis and scientific computing, including works with Roland Glowinski on the numerical analysis of variational inequalities and control theory (1981), and with Olivier Pironneau on the domain decomposition method.

Among Lions’s many works related to applications, in the 1990s, while he was President of the Centre national d’études spatiales (CNES) and President of the Scientific Council of the National Meteorological Office in France, he developed an interest in the mathematical problems of the ocean, atmosphere, and environment. In one of his books on control (1992), he introduced and studied the concept of sentinels for the control and detection of pollution. With two collaborators, Lions wrote a series of eleven articles and a monograph on the mathematical problems raised by the primitive equations of the atmosphere, of the ocean, and of the coupled atmosphere and ocean, and by related asymptotic and numerical issues.

Another massive work in this area is the nine-volume series that he published and edited with Robert Dautray in 1988, *Mathematical Analysis and Numerical Methods for Science and Technology*, which addresses the problems discussed in the classical book of Richard Courant and David Hilbert (*Methods of Mathematical Physics*, vol. 1 and 2, 1953, 1962) in light of the modern methods. This series was translated into English in a six-volume series. Lions also started to publish and edit with Ciarlet the *Handbook for Numerical Analysis* series, which Ciarlet continued to publish by himself after the death of Lions (1990–2002); together they also edited two series of books in applied mathematics.

**Control and Homogenization** . Lions began new directions of research, “System theory” and optimal control, in the late 1960s when he was scientific director at the newly created Institut de Recherche en Informatique et Automatique (Research Institute for Informatics and Automated Systems, or IRIA). He then worked on control theory corresponding to the second part of the title of his chair at the Collège de France. In its general and industrial sense, control theory refers to the procedure by which one determines the best input in a system—e.g., an industrial plant—to obtain the best output: the closest to the desired result in terms of production cost or quality of the product. It can also refer to the description of the states that a given system can or cannot reach. Depending on the nature of the equations governing the system this can result in substantially difficult mathematical problems. Instead of publishing articles, Lions directly published a research monograph on the optimal control of systems governed by partial differential equations (1971). This unique book, originally published in 1968, became the reference book on the subject; like others of his books, it was translated into English, Russian, Japanese, and Chinese. Subsequently Lions considerably developed the subject, writing nine books partly or totally devoted to control theory, which were published between 1968 and 1992. Two of them, in 1978 and 1982, were written with Alain Bensoussan; one was devoted to the applications of variational inequalities to stochastic control, and the other one to impulse control and variational inequalities. In the 1980s Lions was interested in controllability and he introduced the Hilbert Uniqueness Method (HUM), which he developed in a book published in 1988; this was also the topic of his John von Neumann lecture at the Society for Industrial and Applied Mathematics (SIAM) meeting in Boston in 1986.

Another direction of research in the late 1970s and through the 1980s was homogenization. The purpose of this research is the macroscopic description of materials with a complex microscopic structure; PDEs, asymptotics and stochastic analysis are the tools needed here. His first major work appeared in 1978 in a book with Bensoussan and George Papanicolaou. Lions also followed very closely the related work on G and Gamma Convergence of the Italian school around Ennio De Giorgi.

Lions worked in many areas of mathematics, started and developed numerous ideas, theories, and concepts. Relatively few mathematical objects or results bear his name. In fact he did not want mathematical objects or results to be named after him and he often sent signals to that effect to his close collaborators. Instead he was very good at coining appealing names for new mathematical concepts or objects that he introduced or contributed to.

**Scientific Responsibilities and Other Activities** . The above description of Lions’s scientific work does not give a proper idea of the considerable impact of his work, or the tremendous activity behind it: the original courses and lectures he gave, the plenary lectures at major international congresses, or the seminars in small departments (often in developing countries), his frequent travels to distant destinations, the hundreds of pages of faxes that he exchanged weekly with his collaborators (at a time predating e-mail). Lions was also extremely influential with his students. Lions attracted many young people around him, both French and foreign. A very partial list of his graduate students and scientific descendants appears in the list of the Mathematics Genealogy Project of the American Mathematical Society. In 2007, twenty-four students and 709 descendants were mentioned in their database. Lions had at least fifty students for PhD theses, Thèses d’Etat, or Habilitations corresponding to the postdoctoral level. Many of his students became well-established mathematicians; at the end of his life he had scientific descendants of the sixth generation.

Lions also had regular scientific contacts with many high-level scientists worldwide, whom he visited regularly or who visited him in Paris. The regular visitors included Browder, Peter Lax, and Louis Nirenberg from North America; de Giorgi, Magenes, and Guido Stampacchia from Italy; and Sobolev and Vishik from Russia. Magenes recalls that, among Lions’s countless mathematical initiatives, at the end of World War II, he was the first French mathematician (with Schwartz) to reestablish contact with the Italian mathematical community and to visit Italy. This led to the lasting and very active interaction and collaboration with de Giorgi, Magenes, Prodi, and Stampacchia. Lions also contributed to the development of applied mathematics in Spain and India (Bangalore), and was always very generous of his time with young people for correspondence, advising, and visits.

The scientific research of Lions was only part of his work; the other part was his role as manager and consultant, his responsibilities in governmental organizations, and later his role in high-level industrial companies. He seems to be one of very few mathematicians in modern history to have had at the same time an important research activity and important positions in governmental and industrial organizations.

In 1980 the Research Institute for Informatics and Automated Systems (IRIA) became the Institut National de recherche en informatique et automatique (INRIA) and Lions became its first president, a position which he held until 1984. At INRIA, Lions was both the manager and the scientific head of this new institute, which he literally molded. INRIA has played and still plays an important role in the development of computer sciences in France. Lions got involved as much as possible in all of its scientific and organizational aspects.

In 1984 Lions became president of the Centre National d’Etudes Spatiales (CNES), the French space agency. The previous president of CNES, Hubert Curien, himself a physicist who would become the Minister of Research, foresaw the important role that mathematics would play in space research; he asked Lions to accept this responsibility. In this new position, Lions was confronted with new challenges: besides the scientific ones (to supervise works on mathematics, physics, chemistry, and engineering), he went from directing INRIA—a new institute that he fully shaped—to presiding over a large, active, and well-established institution. Furthermore, he was the first mathematician to hold this position.

A new four-year appointment was proposed to him in 1992, but instead Lions retired from CNES, deciding once more to confront new challenges: the industrial world. For many years he had been working on mathematical problems originating from industry; as president of INRIA and CNES he had many contacts with industry. He decided to enter the industry establishment itself, and he became a member of the scientific council or of the board of directors of large industrial groups. He was president of the scientific committees of Pechiney, Gaz de France, Electricité de France, and France Telecom; high-level scientific consultant at Dassault-Aviation and Elf; and he belonged to the board of directors of Dassault-Systems, Pechiney, Compagnie Saint-Gobain, and Thomson Multimedia.

Lions was president of the French Academy of Sciences from 1997 to 1999. He was secretary (1978–1991) and then president (1991–1994) of the International Mathematical Union. He was also member, secretary, or chairman of countless committees and institutions related to research. He was always dedicated in his efforts to help individuals or young groups in isolated places, especially in developing countries.

Lions received many awards and distinctions for all his activity. He was member or foreign member of about twenty academies, including the French Academy of Sciences, the (U.S.) National Academy of Sciences, the American Academy of Arts and Sciences, the Russian Academy of Sciences, and the Third World Academy of Sciences. He received about twenty honorary degrees. Lions was awarded the John von Neumann Prize in 1986, the Japan Prize and the Harvey Prize in 1991, and the Lagrange Prize in 1999, among others. In France, he was named Commandeur de la Légion d’Honneur and Grand Officier dans l’Ordre National du Mérite.

Jacques-Louis Lions was an exceptional person in many respects. He was a charismatic man, generous, very open and accessible, adept at avoiding conflict. One of the most striking aspects of his personality was his long-term vision; he was able to see and get involved in things that only came to fruition five, ten, or twenty years later. He had many good ideas and he had the mathematical talent, the physical strength, and the human abilities to implement them (Temam, 2001).

## BIBLIOGRAPHY

### WORKS BY LIONS

*Quelques méthodes de résolution des problèmes aux limites non linéaires*. Paris: Dunod, 1969.

*Optimal Control of Systems Governed by Partial Differential Equations*. Translated by S. K. Mitter. Berlin: Springer-Verlag, 1971.

With Alain Bensoussan and George Papanicolaou. *Asymptotic Analysis for Periodic Structures*. Amsterdam: North-Holland, 1978.

With Roland Glowinski and Raymond Trémolières. *Numerical analysis of variational inequalities*. Amsterdam: North-Holland, 1981.

With Robert Dautray. *Mathematical Analysis and Numerical Methods for Science and Technology*. 6 vols. Translated by Ian N. Sneddon. Berlin: Springer-Verlag, 1988–1993.

As editor, with Phillipe G. Ciarlet. *Handbook of Numerical Analysis*, 11 vols. Amsterdam: North-Holland, 1990–2002.

*Sentinelles pour les systèmes distribués à données incomplètes* [Sentinels for distributed systems with incomplete data]. *Recherches en Mathématiques Appliquées*, 21. Paris: Masson, 1992.

*Oeuvres choisies de Jacques-Louis Lions*, Vol. I, *Équations aux dérivées partielles. Interpolation;* Vol. II, *Contrôle, Homogénéisation;* Vol. III, *Analyse numérique, Calcul scientifique, Applications*. Edited by Alain Bensoussan, Philippe G. Ciarlet, Roland Glowinski, Roger Temam, François Murat, and Jean-Pierre Puel. Paris: EDP Sciences, 2003.

### OTHER SOURCES

Ciarlet, Philippe G. “Jacques-Louis Lions, 1928–2001.” *Matapli* 55 (2001): 5–16.

Temam, Roger. “Jacques-Louis Lions, 1928–2001.” *SIAM News* 34, no. 6 (2001).

*Roger Temam*

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