Jindrich Necas, honoured by the Order of Merrit of the Czech Republic by Vaclav Havel, President of the Czech Republic on October 28, 1998, Professor of Mathematics at the Charles University in Prague, Presidential Research Professor at Northern Illinois University and Doctor Honoris Causa at the Technical University of Dresden, has been enriching the Czech and world mathematics with his new ideas in the areas of partial differential equations, nonlinear functional analysis and applications of the both disciplines in continuum mechanics and hydrodynamics for more than forty years.

Born in Prague in December 14, 1929, Jindrich Necas spent his youth in the nearby town of Melnik. He studied mathematics at the Faculty of Sciences of Charles University in Prague between 1948 - 1952. After a short period at the Faculty of Civil Engineering of the Czech Technical University he joined the Mathematical Institute of the Czechoslovak Academy of Sciences where he headed the Department of Partial Differential Equation. Since 1977 he has been a member of the staff of the Faculty of Mathematics and Physics of Charles University, being in 1967 - 1971 the head of the Department of Mathematical Analysis, for many years the head of the Department of Mathematical Modelling and an active and distinguished member of the Scientific Council of the Faculty, .

Let us go back to Necas' first steps in mathematical research. He was
the first PhD student of Ivo Babuska, whom he always recalled with
gratitude. As one of his first serious tasks he cooperated in the
preparation of the pioneering monograph * Mathematical Methods of the
Theory of Plane Elasticity * by Babuska, Rektorys and Vycichlo. It
was mechanics which naturally directed him to applications of
mathematics.

This period ended in 1957 with his defence of the dissertation *Solution of
the Biharmonic Problem for Convex Polygons*. His interests gradually
shifted to the functional analytic methods of solutions to partial
differential equations. It was again I. Babuska who oriented him in
this direction, introduced him to S. L. Sobolev and arranged his trip to
Italy. His visits to Italy and France, where he got acquainted with the
renowned schools of M. Picone, G. Fichera, E. Magenes and J. L. Lions,
deeply influenced the second period of Necas' career.

Here we can find the fundamental contributions of Necas to linear
theory: Rellich's identities and inequalities made it possible to prove
the solvability of a wide class of boundary value problems for generalized
data. They are also important for the application of the finite element
method. This period culminated with the monograph *Les methodes
directes en theorie des equations elliptiques*, which became a
standard reference book and found its way into the world mathematical
literature. We have only to regret that it has never been reedited (and translated
into English). Its originality and richness of ideas was more than
sufficient for J. Necas to receive the Doctor of Science degree in
1966.

Without exaggeration, we can consider him the founder of the Czechoslovak school of modern methods of investigation of both boundary and initial value problems for partial differential equations. An excellent teacher, he influenced many students by his enthusiasm, never ceasing work in mathematics, organizing lectures and seminars and supervising many students to their diploma and Ph. D. theses. Let us mention here two series of Summer schools - one devoted to nonlinear partial differential equations and the second interested in the recent results connected with the Navier - Stokes equations. Both of them have had fundamental significance for the development of these areas.

While giving his monograph the final touch, J. Necas already worked on another important research project. He studied and promoted the methods of solving nonlinear problems, and helped numerous young Czechoslovak mathematicians to start their careers in this domain. He also organized many international events and - last but not least - achieved many important results himself.

Nonlinear differential equations naturally lead to the study of nonlinear
functional analysis and thus the monograph *Spectral Analysis of
Nonlinear Operators* appeared in 1973. Among the many outstanding
results let us mention the infinite dimensional version of Sard's
theorem for analytic functionals which makes it possible to prove
denumerability of the spectrum of a nonlinear operator. Theorems of the
type of Fredholm's alternative represent another leading topic. The
choice of the subject was extremely well-timed and many successors were
appearing soon after the book had been published. This interest has not
ceased till now and has resulted in deep and exact conditions of
solvability of nonlinear boundary value problems. Svatopluk Fucik,
who appears as one of the co-authors of the monograph, together
with Jan Kadlec, who worked primarily on problems characteristic for the
previous period, and younger Rudolf Svarc - were amongst the most
talented and promising Necas' students. It is with deep regret
that the premature deaths of all three of them prevented them from gaining the
same kind of international fame as their teacher.

The period of nonlinearities, describing stationary phenomena, probably reached
its summit in the monograph *Introduction to the Theory of
Nonlinear Elliptic Equations*. Before giving an account of the next period,
we must not omit one direction of his interest, namely,
the problem of regularity of solutions to partial differential
equations. If there is a leitmotif that can be heard through all of
Necas' work, it is exactly this problem, closely connected to the
solution of Hilbert's nineteenth problem.

In 1967 Necas published his crucial work in this field, solving the problem of regularity of generalized solutions of elliptic equations of arbitrarily high order with nonlinear growth in a plane domain. His results allow a generalization for solutions to elliptic systems. In 1968 E. De Giorgi, E. Giusti and M. Miranda published counterexamples convincingly demonstrating that analogous theorems on regularity for systems fail to hold in spaces of dimension higher then two.

The series of papers by Necas devoted to regularity in more dimensional domains can be divided into two groups. One of them can be characterized by the effort to find conditions guaranteeing regularity of weak solutions. Here an important result is an equivalent characterization of elliptic systems whose weak solutions are regular. This characterization is based on theorems of Liouville's type. The fact that Necas' method can be applied to the study of regularity of solutions of both elliptic and parabolic systems demonstrates its general character. During this period Necas also collaborated with many mathematicians (M. Giaquinta, B. Kawohl, J. Naumann). The other group of papers is focused on a better understanding of the structure of the singularities for the systems. J. Necas is the author of numerous examples and counterexamples which help to map the situation.

In the next period Necas resumed his study of continuum mechanics.
Again we can distinguish two fundamental areas of his interest. The former concerns
the mechanics of elasto-plastic bodies. J. Necas is the co-author of
monographs *Mathematical Theory of Elastic and Elasto-plastic bodies:
An Introduction* (with I. Hlavacek), *Solution of Variational
Inequalities in Mechanics* (with I. Hlavacek, J. Haslinger a J.
Lovisek). Let us also mention the theory of elastoplastic bodies
admitting plastic flow and reinforcement, as well as the theory of
contact problems with friction. It was J. Polasek who initiated
Necas' interest in transonic flow where he achieved remarkable results by
using the method of entropic compactification and the method of
viscosity. These results raised deep interest of the mathematical
community. Necas published the monograph
*Ecoulement de fluide, compacite par entropie*. In 1986 M. Padula
presented her proof of the global existence of non-steady isothermal
compressible fluids. This article led Necas and Silhavy to
introduce a model of multipolar fluids satisfying the laws of
thermodynamics. In this model the higher order stress tensor and dependence
of this stress on higher order velocity gradients are taking into
account. With this concept the well-posedness of the model was settled.

The most recent considerations of J. Necas were devoted to
classical incompressible
fluids, namely to the Navier-Stokes fluids and to the fluids
with shear and pressure dependent viscosity.
Essentially new existence, uniquenesss and regularity results were given
for space periodic problem and for Dirichlet boundary value problem.
Large time behaviour of solutions was analysed via the concept of short
trajectories. A comprehensive survey of these results can be found in
a monograph *Weak and Measure Valued Solutions to Evolutionary PDE's* (with J.
Malek, M. Rokyta and J. Ruzicka).

The central theme in the mathematical theory of the Navier-Stokes fluids, i.e. the quesion of global existence of uniquely determined solution, has also become central in the research activities of J. Necas in the last decade. Among many results, we wish to mention his contribution to the proof excluding the possibility of constructing a singular solution in the self-similar form proposed by J. Leray in 1934, for the Cauchy problem.

A significant feature of of Necas' scientific work was his intensive and inspiring collaboration with many mathematicians ranging from the youngest to well-known and experienced colleagues from all over the world. Among them (without trying to get a complete list) we would like to mention: H. Bellout, F. Bloom, Ph. Ciarlet, A. Doktor, M. Feistauer, A. Friedman, M. Giaquinta, K. Groeger, Ch. P. Gupta, W. Hao, I. Hlavacek, O. John, R. Kodnar, V. Kondratiev, A. Kufner, Y. C. Kwong, A. Lehtonen, D. M. Lekveishvili, S. Leonardi, P. L. Lions, J. Lovisek, J. Malek, D. Mayer, M. Muller, P. Neittaanmaki, I. Netuka, A. Novotny, O. A. Oleinik, M. Ruzicka, M. Rokyta, T. Roubicek, M. Silhavy, M. Schonbeck, J. Stara, V. Sverak, L. Travnicek.

We will always remember the breadth of his interests and strivings, his encouragement of young people, his never-ending enthusiasm, his deep and lively interest in mathematics. All these features of his personality have attracted students both in Prague and DeKalb and have influenced many world mathematicians.