**From 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 IR**^{n} **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 2**^{nd}**, 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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