Mark Iosifovich Vishik and His Work:
Award Ceremony of the Honorary Doctorate
at the Free University of Berlin

Bernold Fiedler (ed.)
Institut für Mathematik I, Freie Universität Berlin
Arnimallee 2-6, 14195 Berlin, Germany

Mark Vishik and his work Roger Temam Reprinted from Discrete and Continuous Dynamical Systems with permission from the author
	Roger Temam

The name of Mark Vishik is one of the very first names that I heard when I started research in mathematics in 1964. One year before, in 1963, Mark published an article that had a very deep influence on the theory of nonlinear partial differential equations all along the 1960s, although this paper is nearly forgotten by now, and probably very few know about it. Later on I will describe in detail this part of his career that I witnessed during the preparation of my thesis.

Mark Vishik has had a very interesting and rewarding life, but also a very difficult one. He went through many difficult periods but his talent and his kindness attracted respect and sympathy for him, and, for each of the trial times in his life a good fairy came who saved his career and sometimes his life.

Mark Vishik was born on October 19, 1921 in Lvov. A first stroke of destiny hit him at the age of eight when his father passed away. His mother raised him with three other children with loving care and self-denying commitment. Mark retains warm reminiscences of his childhood despite the material difficulties.

From the gymnasium Mark remembers one of his mathematics teachers, Professor Freilich, who followed unconventional teaching patterns requiring the students to find the proofs of theorems and lemmas by themselves. This teaching method may not be appropriate for everyone, but it certainly provides a very stimulating education for a future researcher.

After gymnasium, Mark entered the physics and mathematics faculty of the Lvov University which was then home to the great pre-war mathematics Polish School: Banach, Schauder, Mazur, Orlicz, Steinhaus and others were teaching there. However we are in 1939 and Mark will not stay long at this University. Nevertheless he spends much time in lectures, seminars and in the library, and he stays long enough to decide that he will devote his life to mathematics.

Mark is the only one of his family who survived the Holocaust. During the first days of the German invasion of Poland in World War II, Mark with some students left Lvov. He undertook an impressive trip, alone and hungry, escaping bombing many times. By foot he went to Vinnitsa, 300 km away, then hidden in freight trains he traveled to Kiev, then Krasnodar, then Makhatchkala some fifteen hundred kms away. In Makhatchkala he entered the Makhatchkala Pedagogical Institute from which he soon graduated. Several times he volunteered for the army but was turned down. Instead, after graduation, he was sent with a group of students to the Valley of Sun for harvesting and there he contracted malaria, reoccurences of which still plague him to this day. He was released from the hospital very weak, but the front line was approaching again and, once more, he had to fly for his life; he was carried on a train which took him to Tbilissi.

A positive turn in his fate occurs in Tbilissi where Professors I.N. Vekua and N.I. Muskhelishvili got to know him and became very supportive of him. He was immediately accepted at the University of Tbilissi, granted State Scholarship and given housing. Mark retains a feeling of deep gratitude to the Georgian mathematicians who in fact saved his life.

After graduating in 1943, Mark became graduate student with Professor Vekua at the Mathematics Institute of the Georgian Academy of Sciences. In 1945, Muskhelishvili sends Mark to the Steklov Mathematics Institute in Moscow to continue his graduate studies. In Moscow at the age of 24, he was immediately exposed to the Moscow mathematicians who will deeply influence his research, in particular Sobolev, Petrovskii, Gelfand, Kolmogorov and others. His thesis adviser in Moscow was Lazar Aronovich Lyusternik. Soon after his arrival in 1947, Mark defended his Kandidat thesis. Then in 1951, after a very short period of four years, he defended his Doctorat Thesis, equivalent of the French or German habilitations, this at the very young age of 30.

From 1947 to 1965, Mark was successively assistant, assistant professor and then full professor in the Department of Higher Mathematics of the Moscow Power Engineering Institute. Then in 1965, Petrovskii invited him to join the Department of Differential Equations at the Mathematics and Mechanics Faculty of the Moscow State University where he worked till 1993; during that period he also held a research position at the Institute for Problems in Mechanics of the USSR Academy of Sciences.

Since 1993 he holds a position of principal researcher at the Institute for Problems of Information Transmission of the Russian Academy of Sciences and he is holding the half time position as professor of Moscow State University. Now that traveling has become easier, Mark travels more frequently. We see him much more often in the US, in France, and in Germany where he obtained a Humboldt Award Professorship from 1997 to 2001 which made him, since then, a regular visitor of the Free University of Berlin and of the University of Stuttgart.

The impact of Mark on mathematics has been diverse, prolonged and very extensive. He has written more than 250 articles and several books. There is no way one could properly describe his work in details in a short article; I will just highlight some aspects of his work, in particular those most familiar to me.

However a technical description of his work would not be sufficient to describe his devotion to science and his deep influence. Beside his own research, Mark had and still has many students, many of whom became themselves well established mathematicians, many still around him in Moscow. For many years Mark Vishik has conducted and he still conducts a research seminar during which he proposed to his students and collaborators many open problems. His students tell of the working days with him at his home, during which they worked on these problems. His students recall also that, during the working days, Mark's wife, Asya Moiseevna, entertained them for delicious and enlightening dinners. Asya has been another of the good fairies who have looked after Mark, a very supportive life-long companion, who sailed with him through quiet and through rough waters. Together they had two sons who became mathematicians, Simon from Temple University and Michael from the University of Texas at Austin who works in areas of PDEs at the same time close and very distinct from his father's.

On the occasion of Mark's eightieth birthday, we wish to Mark and Asya and their family many more years of happiness; and to Mark, beside good health, we wish him more students and many more years of fruitful work.

Some aspects of Mark's work

As indicated before, I will just highlight some aspects of Mark's very broad work. A more detailed description of the work of Professor Vishik can be found in the following references:

(i) Mathematics in the USSR during the forty years 1917-1957, Gostekhizdat, Moscow, 1959, vol. II, 138-139.

(ii) Mathematics in the USSR, 1958-1967, Nauka, Moscow, 1969, vol. II, 247-249.

(iii) M.S. Agranovich, I.M. Gelfand, Yu.A. Dubinskii, O.A. Oleinik, S.L. Sobolev and M.A. Shubin, Uspekhi Mat. Nauk, 37:4, 1982, 213-220, Russian Math. Surveys, 37:4, 1982, 174-184.

(iv) Mark Iosifovich Vishik (on his seventy-fifth birthday), Uspekhi Mat. Nauk, 52:4, 1997, 225-232, Russian Math. Surveys, 52-4, 1977, 853-877.

1. Work in linear partial differential equations: the 1950s

S. Sobolev introduced the spaces which bear his name shortly before World War II, and L. Schwartz discovered the theory of distributions shortly after World War II. It was clear that the theory of partial differential equations previously based on the utilization of Hölder spaces and spaces of continuously differentiable functions would have to be fully revisited and further developed using these new powerful tools.

Very young and very early, Mark was very lucky to be exposed to these new developments at the highest level, around Sobolev, Petrovskii and Gelfand. He fully benefited from his teachers and he imbedded himself in the modern theory of PDEs. Parallel developments occurred in a number of places; in particular in France and in Italy around L. Schwartz, J.-L. Lions and the Italian school (E. Magenes and G. Stampacchia and others), in Sweden around L. Gårding and L. Hörmander, in the US around L. Nirenberg, P. Lax, and others.

Particularly noticeable, among the important contributions of Mark, are the following:

• His Kandidat dissertation in 1947 'On the method of orthogonal projections for linear self-adjoint equations'.
• In 1950 a paper 'On general boundary-value problems for elliptic equations', for which he obtained a Prize of the Moscow Mathematical Society.
• In 1951, his Doctoral Dissertation 'On systems of elliptic differential equations and on general boundary-value problems'.

During this period, he gave a general definition of strongly elliptic operators, described the general form of homogeneous boundary conditions -not necessarily local- for a second order elliptic differential operator for well-posedness (solving a problem set by Gelfand); he worked also on nonhomogeneous boundary value problems thus inspiring Lions and Magenes who started then the work which led to their three-volume book; he also started to work on linear time dependent problems.

Another work done during the period of 1957-1960 is the work with his thesis advisor Lazar Aronovich Lyusternik. Mark had many ideas and he did not need much help from his advisor for his theses. However, they eventually collaborated, and they became friends, a friendship which lasted until the end of Lyusternik's life.

2. Work on singular perturbations: the 1960s (1)

M.I. Vishik and L.A. Lyusternik, Translations of A.M.S., vol. 20, 1962, 239-364.

This long article was and still is a reference article on singular perturbations for elliptic and parabolic problems. The other general work on this subject in the context of the modern theory of PDEs is a subsequent volume by J.-L. Lions which appeared in the Springer-Verlag Lecture Notes in Mathematics Series.

Considering the solution of an elliptic boundary value problem with a small parameter

where \(0<\varepsilon <<1\) and where \(L_1\) is of higher order than \(L_0\), \(L_1>>L_0\), the purpose is to study the behavior of \(u_\varepsilon\) as \(\varepsilon \rightarrow 0\). A wealth of asymptotic expansion results were derived in this article. Typically

where the \(w_i\) are the "interior" limits, the \(v_r\) are the boundary layer correctors (singular in the Sobolev spaces), and \(\varepsilon^{m+1}y_m\) is the remainder.

3. Nonlinear elliptic equations monotone in their highest arguments: the 1960s (2)

M.I. Vishik, Systèmes d'équations aux dérivées partielles fortement elliptiques quasi-linéaires sous forme divergente, Troudi Mosk. Mat. Obv., 12; 1963, 125-184.

In this article M.I. Vishik started by himself the theory of monotone operators broadly studied during the 1960's and later on. The prototype problem is now known as the nonlinear Laplacian:

More generally, in the article quoted above, I.M. Vishik considered equations of the for

(l.o.t. : lower order terms).

In (3.1), the associated nonlinear abstract operator \(A\) satisfies the monotony property

More generally in (3.2), Prof. Vishik considered operators \(A\) which are only monotone in their dominant part. This article of Mark is very technical, a tour de force. J.-L. Lions got aware of it, and detecting very early the potential novelty, he studied it and lectured about it in Italy and elsewhere in 1963 and 1964.

Parallel to this an elegant argument of monotony is proposed by G.I. Minty to study monotone integral equations (Monotone (nonlinear) operators in Hilbert spaces, Duke Math. Journal, 29, 1962, 341-346). Later on, it was found that such an argument was already used by R.J. Kachurovski in 1960 and 1966, and by M.M. Vainberg and R.I. Kachnovski in 1959.

F. Browder noticed that Minty's argument can be applied to equations like (3.1) and many others and he developed his results in F. Browder, Nonlinear elliptic boundary value problems, Bull. Amer. Math. Soc., 69, 1963, 862-874, (followed by many other articles in 1965, 1967, 1969 -the latter a review article-).

Finally J. Leray and J.-L. Lions in their only joint paper, fully recovered the results of M.I. Vishik, using the method of Minty and Browder:

J. Leray and J.-L. Lions, Quelques résultats de Vishik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93, 1965, 97-107.

Following these papers, a wealth of papers on monotone operators have appeared in the 1960s and subsequently. Let us also mention the related developments on

• the theory of nonlinear semigroups, and monotone and pseudo-monotone operators (H. Brezis, A. Pazy, and many others),
• the theory of variational inequalities (G. Fichera, J.-L. Lions and G. Stampacchia, and many others).

It is very likely that without the paper of Mark Vishik, and its dissemination by the lectures of J.-L. Lions in Italy and elsewhere, the development of this chapter of mathematics in the 1960s would have been very different.

4. Statistical theory of fluid mechanics: the 1970s

In the 1970s, Prof. Vishik continues to work on nonlinear partial differential equations and he begins to work on the Navier-Stokes equations, and on the statistical theory of turbulence. This new direction of research now reflects the influence of Andrey Nikolaïevich Kolmogorov. It was undertaken in collaboration with A.V. Fursikov, and it led to the reference book

A.V. Fursikov and M.I. Vishik, Mathematical Problems of Statistical Hydrodynamics, Nauka, Moscow, 1980.

The topics studied in this series of articles and in this book include statistical solutions of the stochastically forced Navier-Stokes equations; Reynolds number functional analytic expansion of the solutions, connections with the problem of moments.

Other work done by M. Vishik in the 1970s includes work on degenerate elliptic problems and on pseudo-differential operators (work with Blekher in particular).

Myself in the 1970's, I worked in related but different areas (on the stochastically forced monotone equations (A. Bensoussan and R. Temam, Equations aux dérivées partielles stochastiques (I), Israel J. Math., 11, 1972, 95-129) and Navier-Stokes equations (A. Bensoussan and R. Temam, Equations stochastiques du type Navier-Stokes, J. Funct. Analysis, 13, 1973, 195-222), and on statistical solutions of the Navier-Stokes equations (C. Foias and R. Temam, Homogeneous statistical solutions of Navier-Stokes equations, Indiana Univ. Math. J., 29, 1980, 913-957), (C. Foias and R. Temam, Self-similar universal homogeneous statistical solutions of the Navier-Stokes equations, Comm. Math. Phys., 90, 1983, 187-206)). These common areas of interest generated close interactions between Mark and me in the 1970s, which would amplify in the 1980s and later.

5. Attractors - Dynamical Systems: the 1980s

In the 1980s, Mark undertook work on attractors and dynamical systems mostly in collaboration with Anatoli Babin. Ciprian Foias and myself worked on the same subject at the same time and this led to many fruitful and friendly interactions.

Prof. Vishik's work done in collaboration with Anatoli Babin, led to a series of articles and then to the reference book:

A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, Nauka, Moscow, 1989 (in Russian); and North-Holland, Amsterdam (1992).

A landmark in this work is his article with A.V. Babin giving a lower bound on the dimensions of an attractor:

A.V. Babin and M. Vishik, Attractors of partial differential equations and estimate of their dimension, Uspekhi Mat. Nauk., 38, 1983, 133-187 (in Russian).

Russian Math. Surveys, 38, 1983, 151-213 (in English).

In this article, studying the space periodic incompressible flow in an elongated rectangle of sides \(L\) and \(L\varepsilon , \varepsilon >0\) small, they obtained a lower bound of the dimension of the global attractor \({\cal A},\) bounding it by the dimension of the unstable manifold of a natural and simple stationary solution. They obtained a lower bound of the form

This lower bound is to be compared to the following upper bounds successively established

Finally M. Ziane using very involved arguments of flows in thin domains obtained the following upper bound which matches (4.1) and shows that both bounds are relevant and optimal in some sense

6. Recent and current work : the 1990s and 2000s

Most recent work of Mark in the 1990s and in the 2000s includes an extensive work on nonautonomous dynamical systems with V.V. Chepyzhov, which led to the very recent book:

Attractors for Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, Colloquium Publications, Vol. 49, 2002.

I would like also to mention a work on averaging in dynamical systems with Bernold Fiedler, Quantitative homogenization of analytic semigroups and reaction diffusion equations with Diophantine spatial frequencies, to appear in Advances in Differential Equations.

Of course other articles are on the way!

Continue reading with Mark Vishik's sharing of his thoughts about the sources of his work.