## Obituaries: Jacques-Louis Lions

**July 10, 2001**

Jacques-Louis Lions, 1928-2001

Jacques-Louis Lions was a scientist of remarkable prescience and immense energy, with vision that extended to the development of entire areas of the mathematical sciences. He understood that mathematics could make contributions to science, and worked to see this potential realized. The founder of the French school of applied mathematics, he also had considerable influence on mathematicians and mathematical institutions worldwide. His mathematical legacy includes 20 books and nearly 600 papers. Beyond all his personal achievements, his greatest satisfaction came from the success of his son Pierre-Louis, who was awarded a Fields Medal at the International Congress of Mathematicians in Zurich in 1994.

Jacques-Louis Lions was born and grew up in Grasse, a charming town in the south of France known for the flowers it provides to the French perfume industry. His father was mayor of Grasse for nearly thirty years. Jacques-Louis's wife and lifelong companion Andrée was also born and grew up in the south of France; they met during World War II, when both were active in the Résistance.

After high school, he went to Paris, attending Ecole Normale Supérieure from 1947 to 1950. He then entered the Centre National de la Recherche Scientifique (CNRS), working under the direction of Laurent Schwartz. Schwartz, who had recently completed the work on distribution theory for which he received a Fields Medal in 1950, saw that the theory of partial differential equations would need to be completely revisited in the context of distribution theory. He engaged several of his PhD students in that work, including Bernard Malgrange, François Trèves, and Jacques-Louis Lions. For his PhD, Lions developed the bases of the modern theory of linear elliptic and evolution equations, approaches to these problems that are still in common use today.

**Mathematical Legacy**The mathematical work of J.L. Lions is simultaneously diverse and well unified. His work is accurately described by the title---which he chose---of his chair at the Collège de France: "Analyse Mathématique des Systèmes et de leur Contrôle." The systems he had in mind are those described by linear and nonlinear partial differential equations; by analysis he meant everything from the most abstract existence theorems to approximation and numerical issues and to computer implementations. Control would come later.

Lions was a professor at the famous Collège de France from 1973 until his retirement in 1998; he had taught earlier at the University of Nancy (1954-1962) and the University of Paris (1962-1973). From 1966 to 1986, he was also a part-time professor at Ecole Polytechnique, the alma mater of many of his graduate students.

In the early 1950s, Lions started to develop the building blocks of what would be his "Analyse des Systèmes." He began by addressing, alone or in collaboration, many issues in linear PDE theory and in distribution theory, including work with J. Deny that is still commonly used and quoted. In the late 1950s he began to work in the first two of the areas in which he did major and lasting research.

Under the influence of Jean Leray, whose seminar at the Collège de France he had begun to attend, Lions became interested in nonlinear PDEs, in particular the incompressible Navier-Stokes equations. The mathematical analysis of the Navier-Stokes equations, which had been dormant since the pioneering work of Leray in the 1930s, came back to life in 1951, when Eberhard 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 (elder brother of the current president of the European Union) proved independently the uniqueness of weak solutions in two space dimensions, publishing the result together in 1959. With his deeper understanding of evolution equations, Lions was also able to make Hopf's proof considerably shorter and his result more accessible. In this way, he contributed, with O.A. Ladyzhenskaya, J. Serrin, and others, to the beginning of the modern theory of mathematical fluid dynamics.

The other major direction he undertook at that time was his work with Enrico Magenes on nonhomogeneous boundary value problems, which led to the publication of a three-volume book in 1968. The wealth of results developed in this work necessitated a better understanding and many different characterizations of Sobolev spaces (introduced by Sobolev in the late 1930s), the systematic study of Sobolev spaces with fractional exponents, and the theory of linear elliptic and parabolic equations in such spaces. Developed in parallel was the theory of interpolation between Hilbert or Banach spaces, i.e., the construction of a space intermediate, in the topological and set-inclusion sense, between two given spaces. Lions made substantial contributions to interpolation and in 1953, during a one-year postdoctoral visit with Nachman Aronszajn at the University of Kansas, initiated the interpolation between Hilbert spaces. All this work is very "linear," but its importance and use in nonlinear problems have been considerable.

A new adventure began for Lions in the early 1960s, when he met (in spirit) another of his intellectual mentors, John von Neumann. By then, using computers built from his early designs, von Neumann was developing numerical methods for the solution of PDEs from fluid mechanics and meteorology. At a time when the French mathematical school was almost exclusively engaged in the development of the Bourbaki program, Lions---virtually alone in France---dreamed of an important future for mathematics in these new directions; he threw himself into this new work, while still continuing to produce high-level theoretical work on PDEs.

Without yet publishing in the area, Lions started the French school of numerical analysis. In a loft belonging to the CNRS (Institut Blaise Pascal), away from the (old) Institut Henri Poincaré that was the official center of Paris mathematical life, Lions began to teach a graduate course in numerical analysis. Among the first to receive degrees in this new area were (in chronological order) Jean Céa, Pierre-Arnaud Raviart, and Jean-Pierre Aubin.

At a time when few books devoted to the numerical analysis of PDEs were available, the mimeographed notes from the course at Institut Blaise Pascal were used worldwide to start numerical analysis programs. In his course, Lions used from the beginning the variational theory of boundary value problems that he had developed in his thesis and subsequent work. This point of view was further developed in the first theses that he directed in numerical analysis; the result was an appropriate framework for the development of finite element methods and many subsequent advances in numerical analysis. In this way, Lions also played an important indirect role in research on the numerical analysis of PDEs.

Along with his involvement in numerical analysis, Lions produced, as mentioned earlier, high-level theoretical work. He was involved in the development of the theory of nonlinear equations that are monotone in their highest arguments, a theory based on an idea of George Minty and Felix Browder. His only work with Jean Leray, published in 1965, was one of the most general results in the theory of monotone operators, extending and considerably simplifying an earlier result of Mark Vishik. Lions and Guido Stampacchia published two papers (1965 and 1967) in which they developed the theory of variational inequalities. Lions subsequently continued to develop this theory---alone, with Haim Brezis, and later with George Duvaut in a book devoted to the application of variational inequalities to many concrete and specific problems in continuum mechanics and physics (1972). His theoretical work on nonlinear PDEs is included in a 1969 book that, for the time, was exhaustive. This valuable book is little known (the company that acquired the rights to the English translation having at some point gone bankrupt).

The late 1960s were again a time of new directions for Lions. At that time he was a scientific director at the newly created Institut de Recherche en Informatique et Automatique (IRIA). Alain Bensoussan, and later Roland Glowinski, became his students and then his collaborators at IRIA. His work there included the numerical analysis of variational inequalities, leading to a two-volume book co-authored by Glowinski and R. Tremolières and published in 1976.

The French word "automatique" means automatic control, or more generally control. Control is not usually paired with computer science; its presence in the name of this institute can be attributed to the influence of Pierre Faurre, a younger scientist and industrialist who was highly respected by Lions (and who died a few months before him). At IRIA, then, Lions discovered "system theory," which became a new component of his activity, and he turned his attention to control theory. His first publication in the area, a 1968 research monograph on the optimal control of systems governed by PDEs, became the standard reference on the subject; like many of his other books, it was translated into English, Russian, Japanese, and Chinese. He worked extensively in the area, writing nine books partly or totally devoted to control theory (published between 1968 and 1992). Two of them, in 1978 and 1982, were written with Alain Bensoussan, one devoted to the applications of quasi-variational inequalities to stochastic control and the other to impulse control and quasi-variational inequalities. A new interest for Lions in the 1980s was controllability, and he introduced the Hilbert uniqueness method, the subject of a 1988 book and also of his John von Neumann Lecture at the SIAM meeting in Boston in 1986.

Another research direction in the late 1970s and through the 1980s was homogenization, the macroscopic description of materials with complex microscopic structures via PDEs and asymptotic and stochastic analysis. The first major work of Lions in this area was a 1978 book with Bensoussan and George Papanicolaou. Luc Tartar, one of his former students, continued to develop this subject. Lions also followed very closely the related work of Ennio De Giorgi and the Italian school on G convergence and ? convergence.

In the 1990s, as president of both the Centre National d'Etudes Spatiales (CNES) and the Scientific Council of the National Meteorological Office in France, Lions developed an interest in mathematical problems of the ocean, atmosphere, and environment. The environment was the only subject for which he was willing to get involved in politics; in particular, he was heavily involved in the Venice Environment Initiative Network (VEIN), a project that did not come to maturation. On the scientific side, in one of his books on control, Lions introduced and studied the concept of "sentinels" for the control and detection of pollution. With two collaborators, he wrote 11 articles and a monograph on mathematical problems arising from the equations governing the motion of the atmosphere, the ocean, and the coupled atmosphere-ocean, and on related asymptotic and numerical issues.

More recently, he returned to domain decomposition methods, on which he had written as early as 1972. His most recent work in this area, with Olivier Pironneau, led to nine papers, and a book was in progress.

Another massive work of Lions is *Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques*, the nine-volume series (six volumes in the English translation) that he edited with Robert Dautray (1988) and that is seen by some as a modern version of the Courant and Hilbert book. In the course of this work, as on many other occasions, Dautray was struck by the deep insight of Lions into the physics of the problems, by his ability to raise physically relevant questions not always apparent to the physicists with whom he was interacting. Lions also started to edit, with Philippe G. Ciarlet, the *Handbook for Numerical Analysis* series (Elsevier North-Holland), which Ciarlet intends to bring to completion; together, they also edited two series of books in applied mathematics.

A brief description of the scientific work of Jacques-Louis Lions cannot give a proper idea of the considerable impact of his work, or of the tremendous activity behind it: the original courses and lectures he gave, whether plenary lectures at major international congresses or seminars in small departments, in some cases in developing countries; his frequent travels to distant destinations; the hundreds of pages of faxes that he exchanged weekly with his collaborators.

Nor can a short description do justice to his work with students. Lions attracted many young people, from within and outside France, to work with him. He supervised at least fifty PhD theses (mostly Thèses d'Etat) and Habilitations (at the postdoctoral level). All his students were delighted and amazed at his quick reading of their drafts, and at his availability to each of them. Also a factor in his success as an adviser was his ability to determine very quickly which research would suit a new student, and then to tailor new problems to the student's abilities. Always careful not to influence his students too much, he described himself as a counselor, striving to help each student develop the best of his/her possibilities. Many of his students themselves have become well-established mathematicians; by the end of his life, Jacques-Louis Lions had sixth- generation scientific descendants. Most people working in France on the numerical analysis of PDEs are his scientific descendants, as are a significant part of those working in applied PDEs.

Lions also had regular scientific contacts with high-level scientists worldwide. Many visited him in Paris, usually speaking at one of the seminars he was directing, thus providing up-to-date information for his students and collaborators. For 36 years, Lions directed one or two weekly high-level seminars (one applied and one theoretical from 1962 to 1984). When he left INRIA, the two seminars merged at the Collège de France, and, until 1998, one, two, or three lectures were given at the Friday afternoon seminar. Regular visitors included Felix Browder, Peter Lax, and Louis Nirenberg from the U.S., Shmuel Agmon from Israel, Ennio De Giorgi, Enrico Magenes, and Guido Stam-pacchia from Italy, and Sergei Sobolev and Mark Vishik from the former USSR.

As one among countless initiatives, Enrico Magenes recalls that, at the end of World War II, J.L. Lions was the first French mathematician, with Laurent Schwartz, to reestablish contact with the Italian mathematical community and to visit Italy. From that beginning came lasting and very active interactions, with De Giorgi, Magenes, Prodi, and Stampacchia. He also started to receive and guide a long series of Italian postdoctoral researchers, who eventually became themselves well-established mathematicians, the first one (in Nancy) being Emilio Gagliardo. He also helped in the development of applied mathematics in Spain and India (Bangalore), always especially generous with his time with young people.

**Scientific Responsibilities and Other Activities**The scientific research of Jacques-Louis Lions is only one aspect of his activity-also considerable were his role as manager and consultant, his responsibilities in governmental organizations, and, later, his activities in industry. He seems to have been one of the very few mathematicians in modern history to combine important research activity with high-level positions in government and industry.

In 1980 IRIA became INRIA ("N" is for National), and Lions became its first president, a position he held until 1984. He was both the manager and the scientific head of the new institute, which he literally molded. INRIA has played and continues to play an important role in the development of computer science in France. As far as possible, Lions was involved in all scientific and organizational aspects of the institute.

Lions initiated the expansion of INRIA outside Paris, creating in particular the centers INRIA-Sophia (near Nice) and INRIA-Rennes. He organized the institute around projects, a very efficient approach involving teams with precise objectives, budget assignments, and managerial responsibilities, as well as frequent reports and evaluations. He insisted that a project have three pillars: scientific excellence, relevance to applications, and international cooperation. He created Simulog, the first subsidiary of INRIA, which would be followed by a long series of successful spinoffs. During his four years as president, Lions established the basic principles on which the significant and longstanding success of INRIA has been built.

In 1984, Lions became president of the Centre National d'Etudes Spatiales, the French space agency. The former president, Hubert Curien, a physicist who went on to become minister of research, had foreseen the important role that mathematics would play in space research and asked Lions to accept this responsibility. The new position not only presented Lions with the scientific challenges of supervising work on mathematics, physics, chemistry, and engineer-ing; he was also moving from INRIA---a new institute that he had shaped---to a large, active, and well-established institution. He was, moreover, the first mathematician to hold the position. That he did well is not in doubt: He was reappointed for a second four-year term, and the current president of CNES, Alain Bensoussan, first appointed in 1997, is also a mathematician.

Lions was president of CNES during a period of economic growth that saw the launching of several big programs, including Ariane 4, Ariane 5, and follow-ons of the SPOT series. In style, he very much resembled his predecessor. He was a strong advocate of CNES programmes with the ministries in charge, and, more generally, with politicians, industrialists, and decision- makers. A man of conviction, he put his credibility at stake on these programs and instilled confidence in those who worked with him.

He was particularly effective in three areas. First, with the failure of the V15 Ariane launch in September 1985, he insisted that the engineers invest heavily in modeling and numerical investigations in order to trace the causes of the failure. Later, he initiated a basic research and technology program at CNES, to be carried out in close association with other institutes and research bodies. Finally, he successfully negotiated with NASA to create the French-U.S. space-oceanography program Topex/Poséidon, at a time when NASA payloads on Ariane launch vehicles were banned by the U.S. Congress. He was very active in French-Soviet, and then French-Russian, negotiations to get flight opportunities for French astronauts. Numerous manned space missions, such as Jean-Loup Chrétien's second flight and Michel Tognini's mission, have to be credited to Lions's action.

Among the numerous program decisions made during the presidency of Lions are SPOT 3 (October 1987), SPOT 4 (July 1989), and Ariane 5 (1987). He created several CNES subsidiaries, such as CLS/Argos and Novespace (1986), Scot-Conseil (1987), CERFACS (1988), MEDES (1989), and DERSI (1991).

In 1996 (after his departure from CNES), the new rocket Ariane 501 failed on its maiden flight. Lions was asked to chair the European Space Agency committee created to investigate the failure (a report on the investigations conducted under his direction appeared in SIAM News in October 1996). The failure was traced to numerical overflow, which led to an erroneous orientation of the rocket-at which point it was destroyed by ground command. It is now a textbook example of overflows in computing.

In 1992, declining reappointment, Lions retired from CNES, opting once more for new challenges: the industrial world. He had worked for many years on mathematical problems originating in industry and then, as president of INRIA and CNES, had had many contacts with industry. Now, entering the industrial establishment itself, he soon emerged as president of the scientific committees of Pechiney, Gaz de France, Electricité de France, and France Telecom, a high-level scientific consultant at Dassault-Aviation and Elf, and a member of the board of directors of Dassault-Systems, Pechiney, Compagnie Saint-Gobain, and Thomson Multimedia.

Jacques-Louis 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, initiating during his presidency the 2000 World Mathematical Year. He was also a member, secretary, or chairman of countless committees and institutions in research institutions. He spared no effort to help people just getting started, especially in isolated places and developing countries.

Recognized many times for all his activity, Lions was a member or foreign member of about 20 national academies, including the French Academy of Sciences, the (U.S.) National Academy of Sciences and the American Academy of Arts and Sciences, the Russian Academy of Sciences, and the Third World Academy of Sciences. He received about 20 honorary degrees, and the long list of his other awards includes SIAM's John von Neumann Lecture (1986), the Japan Prize and the Harvey Prize (1991), SIAM's Reid Prize (1998), and the Lagrange Prize (awarded at ICIAM 99, in Edinburgh). In France, he was 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. Charismatic, generous, open, and accessible, he avoided conflict whenever possible. Among the most striking aspects of his personality was his long-term vision; he was able to see and pursue ideas that would come to maturity only five, ten, or twenty years later. He had many good ideas, and he had the mathematical talent, the physical strength, and the understanding of people needed to implement them.

At the celebration of his 60th birthday, Jean Céa wrote that Lions was at the same time a simple and a complex person. He was indeed: Always very kind, he could nevertheless make and stick to difficult decisions when necessary. He had a good sense of humor and was able to inject humor even into very serious matters or difficult situations. He set high standards but was never anything but kind to those who did not meet those standards. His long-term vision would often put him out in front of others, giving him time to elaborate subtle strategies, but he would withdraw to avoid conflict when his suggestions were not adopted.

Despite the many prizes, awards, and distinctions accorded to Jacques-Louis Lions, we believe that he gave far more than he received. He will be very much missed by his friends and colleagues worldwide.

*Roger M. Temam, a student and close collaborator of Jacques-Louis Lions, prepared this appreciation of his life and work at the invitation of SIAM News. He acknowledges the many friends, colleagues, and collaborators of J.L. Lions whose helpful suggestions have been incorporated; Alain Bensoussan, in particular, provided information about the years at INRIA and CNES.*