11. Jacques-Louis Lions e sua influência na matemática do Brasil
Jacques-Louis Lions
and his influence in the Mathematics in Brazil
Luis Adauto Medeiros
Instituto de Matemática –UFRJ
julho de 2001

Jacques-Louis Lions
and his Influence in the Mathematics in Brazil

Professor at "Collège de France", member and past Vice-   President and President of the French Academy of Sciences, member, among several others, of the Brazilian Academy of Sciences, past President of the International Mathematic Union from 1990 to 1994, Jacques- Louis Lions was born in Grasse, France on May 2nd, 1928 and died in Paris on May 17, 2001.
I got in contact with him for the first time in 1965 when I was working with Felix E. Browder in the United States on nonlinear hyperbolic partial differential equations. At that time, motivated in part by the fundamental contributions by Konrad Jörgens [12], Felix E. Browder [6] and Irving Segal [46], that was a flowering research field. Simultaneously, Jacques-Louis Lions and Walter A. Strauss [29] were also developing independently mathematical methods for boundary value problems for nonlinear hyperbolic partial differential equations. In October 27, 1965, he sent me a letter: "Browder showed me a pre-print of your paper on nonlinear wave equations. Could you kindly send me a copy of it, if available?", see [32]. That became the starting point a strong friendship that lasted for 36 years.
By that time I had already read his fascinating paper [16] in which he applied Functional Analysis and Schwartz’s distributions to investigate boundary value problems for partial differential equations. That was precisely the mathematics I wanted to do.
From then on, we frequently exchanged letters on mathematical problems and also about the organization and orientation of Mathematics.
In 1971/72, with the support of Laurent Schwartz and Leopoldo Nachbin, I had the chance of visiting the "Laboratoire d’Analyse Numérique de l’Université de Paris VI", with a scholarship of the French Government. Lions was the Chairman of the Laboratoire at that time and I had the opportunity of meeting him personally and to attend his lectures in Jussieu. The first day I met him he introduced me to E. Magenes and offered me his book on "Les Inéquations en Mécanique et en Physique", in collaboration with G. Duvaut, cf. [9].
From August 15 to 24, 1972 the "Colóquio de Análise" was held at "Instituto de Matemática da Universidade Federal do Rio de Janeiro" (IM-UFRJ) and I invited him to take part in this meeting. He accepted and gave several lectures published in the proceedings, see [41]. This was the first time Lions visited Brazil.
Later on from September 17 to December 15, 1973, I took part in the Summer School on Mathematical and Numerical Methods in Fluid Mechanics, held at Trieste, Italy. Lions was a member of the organizing Committee, with T. B. Benjamin, G. I. Marchuk and G. Stampacchia. During this School, I had the opportunity to discuss with Benjamin about a model in Fluid Mechanics that he had introduced with other coworkers, the by now well known Benjamin-Bona-Mahony (BBM) equation. Coming back to Brazil I worked with my colleagues on

graphicFrom the left to the right: Lions, Antman, Benjamin, Zago and W. Strauss.

that equation and that lead to several publications (see for instance [39] and [37]).
The close connection we had with Lions played a fundamental role in the organization of our research on partial differential equations and control at IM- UFRJ. Initially we were very much inspired by his works: [17], [18], [20], [21] and [28].
Together with Leopoldo Nachbin we organized a work plan so that the researchers of Lions’ School could come to IM-   UFRJ. Those visits played a fundamental role in the development of the program in Analysis, Mechanics and Partial Differential Equations.
Later, in May 1974, I met Lions in New Orleans, USA, in a Summer School on Partial Differential Equations organized by Jerome Goldstein for mathematicians of Latin origin, cf. [11]. Lions and I discussed on non-cylindrical problems for hyperbolic equations, cf. op. cit.. Haïm Brezis gave several lectures on variational inequalities, mainly on the regularity of solutions, cf. [5]. Coming back to Rio de Janeiro, I and my colleagues of IM- UFRJ, started learning about variational inequalities following the papers: [47], [19], [3] and [4]. Later we also obtained unilateral results for Kirchhoff’s operator [38] and for Navier- Stokes equations [48].
In August 1976 the "Colóquio Franco Brasileiro" on numerical methods was held in COPPE- UFRJ. Lobo Carneiro was the Chairman in that occasion and Roland Glowinsky and George Duvaut lectured in the developments and achievements of Lions’ School in France.
By that time our group in IM- UFRJ was already well organized and I proposed to Lions to organize an international colloquium in partial differential equations to be held in Rio de Janeiro. In this way the "International Symposium on Continuous Mechanics and Partial Differential Equations" was held in Rio from August 1 to 5, 1977, cf. [42].
During that meeting, Lions delivered several lectures on various models arising in Mathematical-Physics and, among others, he discussed about the nonlinear Kirchhoff’s equation for the small vibrations of elastic bodies, (cf. [14] pp. 246-284). In 1944, Bernstein [2] had studied the model in one space dimension employing Fourier series and, later, in 1975, Pohozhaev [43] published several results on boundary value problems for the same equation for particular classes of initial data. In his lectures, Lions reformulated the problem, being inspired on its physical interpretation, and treated it by his own methodology. That symposium opened a large field of research topics and activities.
One of the problems that was formulated in that method was an abstract Kirchhoff’s equation in an unbounded domain, where compactness methods could not be employed. If we consider the whole space IRn the Fourier transformation could be applied with success. In a letter dated on December 6, 1977 Lions suggested: "I have the feeling that Fourier has nothing to do with this business... if we use spectral decomposition..." In this way we began working on the diagonalization process of Von Neumann-Diximier and applied it successfully to the nonlinear operator of Kirchhoff type, cf. [31].
From then on, we have had a frequent and intensive exchange of researchers from his School and IM-UFRJ. In particular, several students and professors from IM-UFRJ visited French Universities: Université d’Orsay, Universitéde Paris VI, INRIA, Ecole Polytechnique, Université de Nancy, etc.
The IV Latin American School of Mathematics was held in Lima, Peru in July 1978. As a member of the Organizing Committee, Rivera Rodriguez invited Lions and he lectured on "Some Problems Connected with Navier Stokes Equations" published in the proceedings of the conference. In that occasion we discussed on a model of wave equation with a "pressure" term, see [24], pp. 127 or [27] (dedicated to Louis Nirenberg) and [13].
When we came back from Lima, Lions visited IM-UFRJ and gave us a series of lectures on a model of mixed hyperbolic- parabolic type arising in supersonic fluids. That topic concentrated our attention for some time ( [30] , [33]).
In July 1980, Lions visited IM- UFRJ giving several lectures on Functional Space arising in the theory of control for partial differential equations (see [22]). These methods were later developed [44], [45] and [34].
After that visit to IM-UFRJ, Rivera Rodriguez visited "Collège de France" where he developed research activities with Lions on the control of singular systems. On December 2nd, 1982, he wrote to me: "I am very satisfied with Rivera. He has obtained nice results which permit, I am sure, to treat a lot of cases ( He works on non well set systems and optimal control)". The results Lions was referring to the characterization of a class of convex sets allowing to get optimality system and of a space of ultra distributions for the study of non well posed problems ( [44] and [45]).
In 1986, with surprise, I received from Lions a long manuscript on his Hilbert Uniqueness Method (HUM) for the exact controllability of wave equations, later published in [23]. A working seminar was held at IM-UFRJ to study this work and this lead to several research articles ([1], [35], [40], among others).
In 1990/91, I attended his course at "Collège de France" on "Sentinels", a method he introduced to treat problems of partial differential equation with incomplete data, and that became another topic of interest to the group at IM-UFRJ.
In the nineties I visited Lions in Paris several times. That was always of source of ideas and problems for us. In particular, we worked on approximate controllability following his papers [24], [25] and that by Fabre, Puel and Zuazua [10], and Zuazua [50].
Up to now I have briefly described part of the interaction that Lions had with IM- UFRJ and how this influenced our education and research activities. But the influence of Lions in the Mathematics in Brazil is even richer. First of all, I have to mention that those researchers that were oriented by Lions and his closed collaborators have now themselves oriented many younger researchers and University Professors that are now spread all over Brazil and other countries in Latin America.
The contact with Lions was decisive to open the possibility for several mathematicians of his school coming as visiting professors to Brazil mainly to IM-UFRJ. Among others, I mention:
  • Jean Pierre Puel developed projects with members of IM- UFRJ and open possibilities for our professors to visit "l’ École Polytechnique".
  • Thierry Cazenave comes frequently, as visiting, developing projects with professors of IM- UFRJ and was pos doctor adviser of some of them at "Université Paris VI".
  • Otared Kavian has connected IM-UFRJ with the Project ALFA (Amérique Latine Formation Académique) coordinated by himself, Rolci A . Cipolatti and Flavio Dickstein. This Project opened the possibility to complete the education of ours students and professors.
  • Enrique Zuazua, from "Universidad Complutense de Madrid", comes frequently to Brazil and he was adviser of four thesis of Doctor in Science by UFRJ, three of those partially supported by Project ALFA.
At the institutional level, I had the chance of introducing Lions to the Directors of the "Instituto de Matemática Pura e Aplicada" (IMPA) and "Laboratório Nacional de Computação Científica (LNCC), that, at that time, were within the structure of the "Conselho Nacional de Pesquisas" (CNPq). At that time Djairo Guedes de Figueiredo was President of the "Sociedade Brasileira de Matemática" (SBM) and with his cooperation we organized a visit of Lions to the "Universidade de Brasília" (UnB), "Instituto de Matemática e Estatística - Universidade de São Paulo" (IME-USP), IM- UFRJ and IMPA at Rio de Janeiro-RJ, with the support of SBM.
In 1979 Lions was elected member of "Academia Brasileira de Ciências", (ABC).
In the eighties, Marco Antonio Raupp was appointed director of "Instituto Nacional de Pesquisas Espaciais" (INPE) and, by that time, there was a bilateral agreement "Brazil- France" between INPE and the "Centre National d’Etudes Spatiales" (CNES). Lions was president of CNES at that time and he strongly supported this agreement and reinforced the cooperation of these two institutions, with very profitable consequences for Brazil.
Lions, as a close collaborator and member of the ABC, always supported Brazilian Mathematics in the international scene. From 1990 to 1994, as President of the "International Mathematics Union" (IMU), he had a strong collaboration with Jacob Palis that acted as Secretary of the Union. Jacob Palis is at present Director of IMPA and President of IMU.
Lions’ death leaves a large empty space, but at the same time he has given us a beautiful example of a mathematician who developed a universal art, with the personal print of his own spirit, trying always to apply his creation on all aspects of the life on the Earth.

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