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

 

 

The Sources of My Work Mark Vishik
	Mark Vishik

I am profoundly grateful for being awarded an Honorary Doctorate of the Freie Universität Berlin. I would like to speak about the sources of my mathematical work.

In 1945 academicians Muskhelishvili and Vekua helped me come to Moscow and become a graduate student of the Steklov Mathematical Institute of the Academy of Sciences. Lazar Aronovich Lyusternik was my advisor in Moscow. I was a participant of his seminar. Under his supervision I was free to do what I wanted. At Plesner's seminar, I gave a talk about the famous paper of Herman Weyl about the method of orthogonal projections. Unfortunately, my knowledge of English at that time was almost zero. Therefore I understood only the mathematical formulas and a couple of words in this paper. But the preparation of this talk was very useful for me. I found a generalization of Weyl's method for a general self-adjoint elliptic differential equation of order \(2m\). Then I wrote my Ph.D. (candidate) dissertation based on this work. Lyusternik helped me to write the introduction. I.G. Petrovskii and S.L. Sobolev were the referees of my Ph.D. dissertation. I received my Ph.D. at the Steklov Institute.

For 33 years (from 1945 till 1978) I was a permanent participant of the well-known Gelfand seminar. Gelfand considered all areas of mathematics as connected with each other. All new ideas, new directions, new outstanding mathematical works were reported at this seminar. Gelfand invited the best specialists in the corresponding area and they gave talks at the seminar. But Gelfand often interrupted their presentations, asked questions, discussed, and posed some problems. He asked the lecturer to repeat some parts of the talk and to explain certain details once again. The same happened when one of the permanent participants of the seminar gave a talk on his own new work. A discussion between Gelfand and the speaker about the related problems was the main part of the talk. Gelfand always wanted to verify that everything was reasonable in the work; he posed problems connected with the talk. The seminar often became a creative discussion that was very useful for the participants. Gelfand often asked the participants about their opinion about the talk. Gelfand's seminar has set very high standards for the participants and the seminar extended their mathematical horizon very much. I.M. Gelfand taught the participants right and deep understanding of mathematics.

In 1946 Gelfand organized a small seminar for three participants: O.A. Ladyzhenskaya, O.A. Oleinik and myself. As a result Ladyzhenskaya started working on the problem of describing the domain of the Laplacian with homogeneous Dirichlet boundary conditions in \(L_2(\Omega)\). She proved that the domain is \({\cal D}(A)=H^2 \cap H^1_0. Oleinik worked together with I.G. Petrovskii at that time. I have started working on the problem of describing general boundary conditions for the second-order elliptic differential equation, i.e., when the corresponding operator with these boundary conditions is Fredholm or even an invertible operator.

From the functional analysis point of view this problem is connected with the problem of the extension of operators in a Hilbert space. For example, you have the Laplacian operator defined on the minimal domain \({\cal D}_0(\Delta)\) of functions, which vanish with their first derivatives on the boundary. The maximal Laplacian is defined on all the functions from \(H^2(\Omega)\) without boundary conditions. We look for all extensions of the minimal Laplacian operator which are invertible in \(L_2(\Omega)\) or are Fredholm operators. I proved a general abstract theorem, where necessary and sufficient conditions for the existence of an invertible or a Fredholm extension were given. I found all such extensions for the minimal second-order elliptic differential operators and described their domains by homogeneous boundary conditions. This work was included in my habilitation doctoral thesis. For general differential operators of arbitrary order with constant coefficients, Lars Hörmander proved that the above mentioned necessary and sufficient conditions for existence of an invertible extension of the corresponding minimal operator are fulfilled.

S.L. Sobolev had a great mathematical influence on me. In 1946 at Moscow State University he gave a course on the methods of functional analysis in partial differential equations. This course contained embedding theorems, their applications to the Cauchy problem for hyperbolic equations, their applications to boundary-value problems for polyharmonic equations and other problems. For the Cauchy problem for hyperbolic equations, he first constructed their solutions in \(H^s\) spaces. Then, by means of embedding theorems, he found conditions when the solution belongs to \(C^2\). From here he obtained minimal conditions on the initial data when there exists a classical \(C^2\)-solution. I became an enthusiastic follower of Sobolev's ideas.

Some years later I studied the boundary-value problem for so-called strongly elliptic systems. The leading part of these systems has the form of a sum of a self-adjoint positive symmetric operator and a skew-symmetric operator. I have constructed a weak solution of the Dirichlet problem for strongly-elliptic systems. Louis Nirenberg proved that this weak solution is smooth up to the boundary of the domain provided that this boundary is sufficiently smooth.

In 1956 we wrote with S.L. Sobolev an article in Doklady of Academy of Sciences about the solutions of non-homogeneous boundary-value problems of elliptic equations belonging to some distributional class. So, for example, we considered the Neumann problem for the Laplace equation with a given measure on the boundary. The measure of a set on the boundary is equal to the integral of the normal derivative of the sought solution over this set on the boundary. The solution of this generalized Neumann problem was found by using the duality between this problem and the Dirichlet problem with corresponding smooth boundary value conditions. This paper has had far-going generalizations. Jaques Lions and Enrico Magenes wrote a book of three volumes entitled 'Non-homogeneous Boundary Value Problems and Applications', where, in particular, a theory of general non-homogeneous problems in various classes of distributions was developed in great detail.

In the forties I.G. Petrovskii organized a seminar on partial differential equations for graduate students and for students of the 4-th and 5-th years. He also had a larger seminar at the chair of differential equations. I often gave talks at these seminars. Usually after the talk Petrovskii asked the speakers to formulate the main result once more. He asked whether all the supposed conditions were really necessary. He very much enjoyed applied mathematical talks. Petrovskii had a deep influence on me as a great mathematician and a great personality. In 1965 he invited me to be a professor at his chair at Moscow State University.

In 1956 we began to work with Lazar Aronovich Lyusternik on the asymptotic behaviour of solutions of equations having a small parameter in the higher order terms. For example we studied the 4-th order equations with left-hand side \(\varepsilon \bigtriangleup^2 -\bigtriangleup\). As \(\varepsilon \to 0+\) we have a plate equation degenerating into a membrane equation. The fourth order elliptic operator requires two boundary conditions, for example, the Dirichlet condition and the normal derivative on the boundary. The limiting membrane equation requires only the Dirichlet boundary condition. Therefore we lose one boundary condition for \(\varepsilon=0\). We proved that the solution of the 4-th order equation can be asymptotically represented as a sum of a solution of the limiting Dirichlet membrane boundary-value problem and a boundary layer function. The boundary layer decreases exponentially along the normals to the boundary and has the form \(\varepsilon e^{-\lambda{n}/{\varepsilon}}\psi(s)\), where \(n\) is the distance to the boundary along the normal vector. The boundary layer function satisfies a certain ordinary differential equation. The main idea of the construction was based on the hypothesis that the boundary layer function changes along the normal direction much faster than in any direction tangent to the boundary. To obtain an asymptotic decomposition of the solution for the equation with small parameter in the higher order terms we constructed two iteration processes. The first one corresponds to the behaviour of the solution inside the domain and the second one corresponds to its behaviour near the boundary. On each iteration step we tried to improve our approximation of the given boundary conditions.

We also studied asymptotic expansions of the eigenvalues and eigenfunctions of elliptic equations with small parameter in the highest derivatives.

For evolution equations with rapidly oscillating boundary values, we obtained a skin-effect first studied by Riemann. We also studied many other singularly perturbed problems.

The collaboration with such a great mathematician as Lyusternik has had a deep influence on me. For five years we have usually been working two days a week, from morning till night. We wrote 25 articles and 3 surveys in the journal Uspekhi (Surveys). During our collaboration I learnt the style of working of a great mathematician, the broadness of his interests. We worked with enthusiasm. Sometimes Lyusternik called me at 2 o'clock in the morning, when he found some useful remarks to our work. The methods developed with Lyusternik were used by many scientists in applied mathematics, mechanics, and physics.

The breaks in our working sessions were very interesting. Lyusternik was a cultured man. He knew many verses of Pushkin, Baratynskii, Tyutchev, Pasternak and other poets by heart. Lyusternik was also a poet himself. He wrote humoristic verses about some professors of mathematics. He was a person with a great sense of humor. Five years of collaboration with such a person have had a deep influence on me. After our collaboration with Lyusternik we and our families became close friends for all future time.

In the beginning of the sixties he proposed to write a book with me on boundary layer problems. Unfortunately, I could not do that because at that time I worked entirely on monotone elliptic operators. In the fifties I had tried to construct a solution of the Dirichlet problem for the non-linear analogue of the Laplacian: \(\sum\frac{\partial}{\partial x_i} \left(\frac{\partial u}{\partial x_i} \right)^p=h\) (where \(p\) is an odd number for simplicity). In the beginning of the sixties I noticed that Galerkin approximations for this equation had at least one solution in a sufficiently large ball. This fact follows from a simple topological argument. By a priori estimates for the Galerkin approximations and the compactness of the approximating solutions I obtained the solution for the non-linear analogue of the Laplacian. Then I constructed solutions for the general nonlinear strongly elliptic systems. Later these systems were called monotone elliptic. I also constructed solutions for general parabolic systems of equations with strongly elliptic right hand side. Many mathematicians worked in this area: F. Browder, H. Brezis, H. Gajewski, J. Leray, J.-L. Lions, Yu.A. Dubinskii, I.V. Skrypnik and many other mathematicians.

In the beginning of the fifties I worked at the Moscow Energy University. I helped to organize the department of Mathematics and a scientific seminar on differential equations at this University. Twelve participants of this seminar received a Ph.D. degree and Dubinskii obtained a habilitation Ph.D.

In the beginning of the sixties I.M. Gelfand proposed that I organize a seminar on differential equations and their applications at Moscow State University. Many young talented mathematicians were participants of this seminar. They worked on various problems in partial differential equations, on spectral theory, on the Fredholm index of elliptic operators and on many other problems. My students and graduate students, some students of Olga Oleinik and Georgi Shilov, and many other mathematicians from different Moscow institutes participated in this seminar; there were about 50 participants. The seminar was very useful for me and, I believe, also for the participants. We reviewed and discussed new interesting articles of such outstanding mathematicians as de Giorgi, L. Hörmander, L. Nirenberg, P. Lax, J. Moser, J.-L. Lions, R. Temam, H. Brezis, L. Amerio, E. Magenes, and many other mathematicians. The participants gave talks about their new results. The seminar is now working permanently for over 40 years. I wrote many joint papers with participants of the seminar.

In the sixties we wrote some papers with M.S. Agranovich about general boundary value problems for parabolic equations of arbitrary order. First it was necessary to study the corresponding general elliptic operator with a large parameter in a sector. This part was very close to the earlier papers of Agmon and Nirenberg. I worked together with Grigorii Eskin for some years. We studied problems for elliptic pseudodifferential equations in a bounded domain. Then G. Eskin wrote a book about these and other problems. The collaboration with Agranovich and Eskin was very useful for me. I learnt a lot from them.

In the beginning of the seventies we organized a small seminar to study statistical solutions of evolution equations. Eberhard Hopf was the first to give a statistical approach for the study of the Cauchy problem for the Navier-Stokes systems, in his famous paper. His approach was developed by Foias, Bensoussan, Temam, Fursikov, and myself and other mathematicians.

Let me briefly describe the statistical approach to the Cauchy problem for the \(2D\) Navier-Stokes system.

Let a measure \(\mu_0(du)\) be given at \(t=0\) on a function space, say, \(L_2(\Omega)\). Here \(\mu_0(du)\) is the probability of the event that the initial point \(u(x)\) is in the domain \(du\) in \(L_2(\Omega)\). The statistical solution corresponding to \(\mu_0(du)\) is the family of measures \(\mu(t,du)\), \(t>0\), which is equal to the evolution of \(\mu_0(du)\) along the trajectories. The characteristic functional \(\chi(t,v)\) of this measure \(\mu(t,du)\) is equal to the Fourier transform of the measure \(\mu(t,du)\). The characteristic functional \(\chi(t,v)\) satisfies the famous Hopf evolution equation with variational derivatives in \(v\). Andrei Fursikov and I proved that if the initial measure \(\mu_0(du)\) is supported in a sufficiently small ball, then the Cauchy problem for the Hopf equation possesses a solution \(\chi(t,v)\), globally in time, which is functionally analytic with respect to \(v\). From this theorem we deduced that the Friedman-Keller infinite chain of equations for the moments of the statistical solution \(\mu(t,du)\) possesses a global solution with respect to \(t\). In the general case, when the initial measure \(\mu_0(du)\) has an arbitrary support, using another method we proved with Fursikov the existence theorem of the Cauchy problem for the Friedman-Keller infinite chain of equations corresponding to the \(3D\) Navier-Stokes equations. Fursikov and I thus constructed a statistical solution for the Cauchy problem for the \(3D\) Navier-Stokes system.

The ideas of Andrei Nikolaevich Kolmogorov on homogeneous turbulence had a deep influence on our later work on statistical solutions. Fursikov and I proved the existence of a homogeneous statistical solution for the Navier-Stokes system which corresponds to a given initial homogeneous measure \(\mu_0(du)\). The homogeneity of the measure is understood with respect to the space variables \(x\). Our work stimulated Kolmogorov to give his last talk at the Moscow Mathematical Society on the turbulence problems which are to be solved. Some of these problems were solved in our book with Fursikov 'Mathematical Problems of Statistical Hydromechanics'. In the Appendix to this book written by Alexander Komech and myself we considered stochastic problems for the Navier-Stokes system and solved some of Kolmogorov's problems. But many of them still remain open.

From the beginning of the eighties, Anatolii Babin and I began to study the global attractors of differential equations and systems of mathematical physics. The theory of attractors for ordinary differential equations and some functional equations had been developed to some extent. The attractors for partial differential equations were systematically studied from the eighties on. Important contributions to this area were made by R. Temam, C. Foias, P. Constantin, J. Hale, G. Sell, A. Haraux, and others. Naturally, our works with Anatolii Babin were influenced by the investigations of these and other mathematicians. We studied and constructed the global attractors for the \(2D\) Navier-Stokes system, the dissipative wave equation, some classes of reaction-diffusion systems and some other equations. When the corresponding evolution equation possesses a global Lyapunov functional we investigated the structure of the attractor and introduced the notion of a regular attractor. Under some additional conditions we proved the exponential attraction property of the global attractor. We also studied the Hausdorff dimension of the attractors. For some examples of evolution equations we gave a lower bound on the dimension of the global attractor. Our work in this area was summarized in our book with Babin 'Attractors of Evolution Equations'.

Since 1993 I am working as a leading scientific researcher at the Institute of Information Transmission Problems, Russian Academy of Sciences. There I have nice conditions for scientific work. In particular, V. Chepyzhov and I have recently written the monograph 'Attractors for Equations of Mathematical Physics', where we exposed our works of the last ten years. I am still a professor at Moscow State University.

With Vladimir Chepyzhov we study the global attractors for nonautonomous evolution equations with some coefficients and forcing terms depending on \(t\). We constructed and studied global and trajectory attractors for the \(3D\) and \(2D\) Navier-Stokes systems with forcing term depending on \(t\), for some classes of non-autonomous reaction-diffusion systems, for the non-autonomous Ginzburg-Landau equation, and for the non-autonomous dissipative wave equation. The Hausdorff and fractal dimension of the global attractor for non-autonomous equations is often infinite. Chepyzhov and I studied the Kolmogorov \(\varepsilon\)-entropy of global attractors for these equations, which is always finite for any \(\varepsilon >0\). We found an upper estimate for the Kolmogorov \(\varepsilon\)-entropy of the global attractor for some classes of non-autonomous equations and systems.

Recently Chepyzhov and I constructed the global attractor for the \(3D\) Navier-Stokes system which has properties analogous to the global attractor for the \(2D\) Navier-Stokes system. We also studied perturbation problems for global attractors, including perturbations by terms which oscillate rapidly in \(x\) and \(t\).

Summarizing, I can say the following. My great teachers from the Moscow State University are one of the main sources of my mathematical work. With some of them I collaborated and published joint articles. I can recall the words by Isaac Newton 'In our work we stand on the shoulders of giants.'

My seminar and its participants have a great influence on me. With many of the participants I collaborated personally. This seminar is an important source of my mathematical work. I am very satisfied that I had many talented graduate students. Many of them have grown into outstanding mathematicians. I learnt very much from them. They are also one of the sources of my work.

I interacted with many mathematicians of other countries. I interacted with many outstanding mathematician from France, USA, Germany, Italy, Sweden and other countries. The interaction with these mathematicians is an important source of my mathematical work.

Recently as Alexander von Humboldt Awardee I had a possibility to work in Germany. I collaborated with mathematicians of the Freie Universität Berlin, mostly with Bernold Fiedler and his colleagues. With Fiedler we wrote several joint papers. I also collaborated with mathematicians of the University of Stuttgart, mostly with Wolfgang Wendland and his mathematical group. I was in Leipzig many times, as a visiting professor of the University of Leipzig. Recently I visited the Max-Planck Institute für Naturwissenschaften in Leipzig, where I collaborated with many young talented mathematicians.

Many thanks to all these people who stimulated my scientific work.

All my conscious life I am fascinated with mathematics, I like to study it. Almost all the time I think about mathematical problems, about what would be useful to do for the development of mathematics. In my life nothing else attracts me as much as this area. Such attractiveness of mathematics for me is undoubtedly one of the important source of my mathematical works.

At home my wife Asya organizes nice conditions for my mathematical work. Many thanks to her.

I am grateful to the speakers and the participants of the Symposium on Partial Differential Equations in my honor. I am grateful to Bernold Fiedler who organized the symposium.

I am profoundly grateful for the Honorary Doctorate of the Freie Universität Berlin.

Many thanks to the participants of this meeting.

Thank you very much.

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Continue by reading the closing remarks.