Pierre-Louis Lions is the son of the famous mathematician Jacques-Louis Lions and Andrée Olivier. He was born in Grasse, Alpes-Maritimes in the Provence- Alpes- Côte- d'Azur region of France, northwest of Cannes. It was the birthplace of his father and the town the family considered home, although at the time of his birth his father was a professor at the University of Nancy. Let us note that Grasse is not very far from Draguignan where Alain Connes, who won a Fields Medal 12 years before Lions won his Fields Medal, was born.

When Pierre-Louis was six years old his father became a professor in Paris and the family lived there. He attended the Lycée Pasteur and then the Lycée Louis-le-Grand before entering the École Normale Supérieure in 1975. Lions studied at the École Normale Supérieure from 1975 to 1979. His thesis, supervised by H Brézis, was presented to the University of Pierre and Marie Curie (formally Paris VI when the University of Paris was split into thirteen separate universities in 1970) and in 1979 he received his Doctorat d'Etat es sciences. On 1 December 1979, after he received his doctorate, Lions married Lila Laurenti. They have one child Dorian.

From 1979 to 1981 Lions held a research post at the Centre National de la Recherche Scientifique in Paris. Then, in 1981, he was appointed professor at the University of Paris-Dauphine. While still holding this post he was attached to the Centre National de la Recherche Scientifique as Director of Research in 1995. He has also held the position of Professor of Applied Mathematics at the École Polytechnique from 1992.

Lions has made some of the most important contributions to the theory of nonlinear partial differential equations through the 1980s and 1990s. Evans, in [2], writes:-

He has made truly fundamental discoveries cutting across many disciplines, pure and applied, and his publications are so numerous and varied as to defy easy classification. Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probalistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods. Pierre-Louis Lions is unique in his unbelievable ability to transcend these boundaries and to solve pressing problems throughout the field.

The references quoted [2], [3] and [4] decribe some important aspects of Lions work which led to the award of a Fields Medal at the International Congress of Mathematicians in Zürich in 1994. The first area of Lions work that is highlighted by both [1] and [3] is his work on "viscosity solutions" for nonlinear partial differential equations. The method was first introduced by Lions in joint work with M G Crandall in 1983 in which they studied Hamilton-Jacobi equations. Lions and others have since applied the method to a wide class of partial differential equations, the so-called "fully nonlinear second order degenerate elliptic partial differential equations." The problem that arises is decribed in [2]:-

... such nonlinear partial differential equation simply do not have smooth or even C^{1}solutions existing after short times. ... The only option is therefore to search for some kind of "weak" solution. This undertaking is in effect to figure out how to allow for certain kinds of "physically correct" singularities and how to forbid others. ... Lions and Crandall at last broke open the problem by focusing attention on viscosity solutions, which are defined in terms of certain inequalities holding wherever the graph of the solution is touched on one side or the other by a smooth test function.

Another equally innovative piece of work by Lions was his work on the Boltzmann equation and other kinetic equations. The Boltzmann equation keeps track of interactions between colliding particles, not individually but in terms of a density. In 1989 Lions, in joint work with DiPerma, was the first to give a rigorous solution with arbitrary initial data.

Another major contribution by Lions, in a long series of important papers, is to variational problems. Varadhan, speaking at the Congress of Mathematicians in Zürich in 1994 about Lions' work [4], said:-

There are many nonlinear PDEs that are Euler equations for variational problems. The first step in solving such equations by the variational method is to show that the extremum is attained. This requires some coercivity or compactness. If the quantity to be minimised has an "energy"-like term involving derivatives, then one has control on local regularity along a minimising sequence.

Lions's clever idea was to introduce "concentration compactness" techniques which look at energy concentrations and so avoid problems which occur when examining the minimising sequences without compactness. He introduced certain measures to handle the concentrations.

Lions has received many awards for his outstanding contributions to mathematics. He is a member of the French Academy of Sciences and he was awarded prizes by the Academy, the Doistau-Blutet Foundation Prize in 1986 and the Ampère Prize in 1992. He also received the IBM Prize in 1987 and the Philip Morris Prize in 1991.

In addition to the Paris Academy, Lions has been elected a member of the Naples Academy and the European Academy. He is also Chevalier of the Légion d'Honneur. He has been awarded an honorary doctorate from Heriot-Watt University in Edinburgh, Scotland. He is on the editorial board of around 25 journals world-wide.

In 2003 three volumes of 'Selected works of Jacques-Louis Lions' were published. The first volume covered his work on Partial differential equations and Interpolation, the second volume contained Control and Homogenization, and the third volume Numerical analysis, Scientific computation and Applications. All three volumes contain a Preface which gives a good overview of his contributions. We quote from the Preface to the third volume:-

Starting in1990, Lions expressed interest in a major application, climatology. The models used in that field consist of complex sets of partial differential equations, including the Navier-Stokes equations and the equations of thermodynamics. ... In spite of what Lions himself liked to call the 'truly diabolical' complexity of the set of partial differential equations, boundary conditions, transmission conditions, nonlinearities, physical hypotheses, etc., that appeared in those models, Lions, in collaboration with Roger Temam and Shou Hong Wang, was able to study the questions of the existence and uniqueness of solutions, to establish the existence of attractors, and to do a numerical analysis of these models. He was even able to teach this material in the classroom, which was quite a pedagogical challenge! In a series of works begun with Évariste Sanchez-Palencia in1995, he also developed the theory of 'sensitive problems', particularly as they arise in the theory of elastic shells....

The variety of topics that Lions tackled in the above works is very impressive. In a long series of notes published in the Comptes Rendus until 2001, Lions returned to numerical analysis, and in particular to parallel computation and domain decomposition methods. These subjects had long interested him ... Always in search of new subjects, Lions even pursued the study of a problem of dislocation in crystallography, a problem that no one really knew how to tackle. And, amazingly enough, he was the first person to establish (in 2000) a result of existence and uniqueness of the solution of this type of problem. Finally, with Vivette Girault, he worked until January 2001 on perfecting a finite element method using two meshes, one 'rough' and one 'fine', for the numerical simulation of the Navier-Stokes equations. The inventiveness and mathematical richness of the article that resulted from that collaboration are remarkable.

One cannot help being struck by the quality, diversity and novelty of the mathematics used in this immense body of work, and by Lions's ability to decipher among the applications some vast areas that had been thought to be inaccessible. Like John von Neumann, for whom he expressed profound admiration, Jacques-Louis Lions was a visionary who understood very quickly that the use of increasingly powerful computing tools could revolutionize the modelling of phenomena and improve our knowledge and mastery of the physical world as long as the corresponding mathematics was created and developed. This was the task to which he devoted himself so admirably.

Finally let us note that Lions was awarded the Prix Thomson (2004) and, on behalf of his team, the Prix Institut de Finance Europlace (2003).

Lions lists his hobbies as cinema and reading, and his favourite sports as rugby and swimming.

**Article by:** *J J O'Connor* and *E F Robertson*

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