Olga Arsenevna Oleı˘nik (1925–2001)

Willi Jäger, Peter Lax, and Cathleen Synge Morawetz

Her Life1 highest-order terms, heralded some of the under- On October 13, 2001, Olga Arsenevna Oleı˘nik, one lying approaches to much of her future work. of a handful of truly exceptional women mathe- Although she also made contributions to algebraic maticians of the twentieth century, died in Moscow geometry, to Hilbert’s sixteenth problem, and to at the age of seventy-six, succumbing, after a long other topics, her forte and lifetime contributions struggle, to cancer. were focused on PDEs that arise in very important Olga Oleı˘nik was born in the Ukraine on July 2, applications such as boundary-layer theory and 1925. In 1941 the Ukraine was invaded by Germany, elasticity. In fact, she constantly emphasized the and the machine factory where Olga’s father was role of PDEs in applications [O1]. bookkeeper was evacuated to Perm in the Urals. The She remained a devoted student and disciple of sixteen-year-old Olga accompanied him and finished Petrovskiı˘ and his memory for her whole life. She high school there. Her mother, her sister, and her succeeded him to the chair of differential equations, nephew remained in the Ukraine. Olga then attended and despite the political difficulties of the seven- the University of Perm, to which the mathematics ties and eighties, she steered the department for- and mechanics faculty of the Moscow State Univer- ward successfully throughout her tenure. Oleı˘nik sity had also been evacuated. There her talents came received many prizes and awards in her lifetime;  to the attention of professors Sof ya Yanovskaya she was named a member of the Russian Academy and Dynnikov, who arranged in 1944 for her to be- of Sciences in 1990 but had earlier been made an come a student in Moscow at the university. She was honorary member of too many foreign academies married briefly and had one son, to whom she was and societies to list here. She was also awarded devoted and who predeceased her. many honorary degrees as well as prizes both in Oleı˘nik continued at the university, receiving her her native land and abroad. In all, she published doctor’s degree in 1954 with a thesis in partial dif- more than three hundred papers and eight mono- ferential equations (PDEs) under the guidance of graphs. I. G. Petrovskiı˘. The topic, partial differential equa- She was deeply involved in improving and ex- tions with small coefficients multiplying the tending contacts between Russian and Western scientists. She was particularly eager to bring the Willi Jäger (WJ) is a professor in the Institute for Applied Western world to Russian mathematicians and was Mathematics at the University of Heidelberg, Germany. His email address is [email protected]. therefore anxious to see Western mathematical lit- erature translated into Russian. An early beneficiary Peter Lax (PL) is professor emeritus at the Courant Institute, New York University. His email address is of such a translation was the new edition of [email protected]. Courant-Hilbert, Vol. 2, which appeared in Russian Cathleen Synge Morawetz (CSM) is professor emeritus in 1962, the same year it appeared in English. When at the Courant Institute, New York University. Her email Courant died in 1972, Oleı˘nik and Paul Aleksandrov address is [email protected]. wrote a long obituary in Uspekhi2 emphasizing 1CSM. The author (CSM) is indebted to Tatiana A. Sha- poshnikova of Moscow University for much of this infor- 2P. S. Aleksandrov and O. A. Oleı˘nik, Uspekhi Mat. Nauk mation. 30 (1975), no. 4 (184), 205–26; MR 52 #7786.

220 NOTICES OF THE AMS VOLUME 50, NUMBER 2 Courant’s contributions to American mathemat- were involved in it. It would ics through his institute at New York University. make this article too tech- Oleı˘nik loved to travel. In 1960 she met Courant nical to describe Oleı˘nik’s and Lax in Moscow on their visit to Alexandrov, most significant achieve- an old friend of Courant’s. Thus when she was in- ments in this area, and we vited to a women’s congress in California shortly will confine the scientific afterwards, she was able to accept Courant’s invi- description to some of the tation to visit New York. It is hard now to remem- many other areas to which ber how unusual such a visit was in the decade she made even more signif- following Stalin’s death. She retained an abiding icant contributions. affection and interest in those working in Courant’s group: K. O. Friedrichs, F. John, P. D. Lax, and Nonlinear Hyperbolic 3 L. Nirenberg. I met her then for the first time, but Equations I became close to her only when, at G. Fichera’s in- During the years 1954–61, vitation, she and I spent a month in 1965 together Olga Oleı˘nik studied the at the University of Rome. The university was theory of nonlinear hyper- “under siege” by the student body, and it was only bolic conservation laws and Fichera’s political skill that made it possible for the propagation of shock Oleı˘nik to give her talks. We were excluded from waves. Her contributions the campus much of the time, so together we hap- were basic and extremely pily toured the wonderful sights of Rome instead. original. Her 1957 paper in Olga Oleı˘nik. Oleı˘nik was a very private person. In the latter the Uspekhi [O2] was par- part of her life she suffered badly from knee trou- ticularly influential. The starting point of her work, ble, which prevented her from walking properly and like much in this field, was Eberhard Hopf’s fun- sometimes kept her hospitalized. During one such damental paper of 1950 [H]. long stay she kept occupied by writing another Oleı˘nik’s results deal mainly with the existence, book. For a woman who had been so active as a uniqueness, and properties of solutions of the mathematician and with her students (she super- initial-value problem for single conservation vised fifty-six dissertations), these spells of inac- laws. She showed that solutions satisfy one-sided tivity were trying. She told me once that as a young Lipschitz conditions and formulated for flux student she had worked in what we would call a functions that have points of inflection what today lumber camp and how much she, in contrast to the is called the Oleı˘nik entropy condition. other girls, had enjoyed doing the hard physical Her principal tool was the parabolic perturba- work. tion of a conservation law. She proved that as the The life span of Olga Oleı˘nik was from the early coefficient of viscosity tends to zero, the solution days of the sovietization of Russia to the complete of the initial-value problem for the parabolic equation tends to a solution, in the sense of collapse of communism. I never heard her express distributions, of the conservation law, and that a political opinion nor make a complaint about this limit satisfies the Oleı˘nik entropy condition. how things were going. She clearly tried to live In 1957 Oleı˘nik and Vvedenskaya considered a within the system she had been raised in, and her discretized form of a conservation law and proved model was always her teacher, Petrovskiı˘. Uprooted the convergence of their solutions to solutions from the Ukraine by the war, she identified herself of the conservation law, as the discretization as a Russian, not as a Ukrainian. Of one thing I am parameter tends to zero [OV]. This provided sure: she believed a larger picture of her country another approach to the problem. was the right one. Oleı˘nik also studied special systems of pairs of Oleı˘nik was devoted to mathematics. She drove conservation laws, which are derivable from second- her theorems to their absolute limits. She never order equations; in 1966 she proved uniqueness of seemed happier than when she was doing mathe- the initial-value problem in the class of solutions matics or working with her students, the most im- that satisfy a one-sided Lipschitz condition [O4]. portant elements in her life. She left a great deal of unfinished work, although she had published so Boundary Layer Theory4 much. We may hope that this work will be com- In their simplest form, Prandtl’s steady 2D bound- pleted by her students and colleagues. ary-layer equations are obtained from the Navier- Oleı˘nik was one of the major figures in the Stokes equations by stretching the variables study, during the fifties and sixties, of elliptic appropriately. If x is distance along the boundary, and parabolic equations. This study was the major field of partial differential equations at the time, and many mathematicians such as Agmon, John, 3PL. Ladyzhenskaya, Morrey, Nirenberg, and Vishik 4CSM.

FEBRUARY 2003 NOTICES OF THE AMS 221 and y, a depth variable, is distance from the so that the equation remains elliptic and to obtain boundary, the equations become after rescaling: estimates for the limiting case when this term vanishes. By 1977, however, Oleı˘nik was directly ux + vy = 0 (mass), using a priori energy estimates [O7]. uux + vuy = −px + µuxx (x-momentum), 6 py =0 (y-momentum). Homogenization of Differential Equations Oleı˘nik’s contributions to homogenization and her The standard boundary conditions are u = v =0on impact on this field were very broad. Mathemati- y =0 and u → U(x) as y →∞. The function U is cal modelling leads naturally to systems with mul- the speed of the flow “along the body” given by tiple scales, for example in electromagnetism, me- inviscid incompressible flow and is related to the chanics, material science, and flows through porous pressure by Bernoulli’s law, media or biological tissues. The resulting underlying structures may be highly oscillatory in space and p + 1 U2 = constant. 2 time. To analyze the transition between the dif- Upstream, say x =0, the incoming flow is pre- ferent scales and to derive effective model equa- scribed. Experimental observations confirm the tions are great challenges to mathematical research scaling. and imply many practical consequences. The main Thus we have a nonlinear parabolic equation for problem lies in identifying the proper scaling and, u with the role of time derivative played by the if possible, deriving effective equations for limits Lagrangian along the particle path. The coefficient when a “scale” goes to 0 or ∞. The random situa- v is to be found from the mass equation. tion especially, the most important in real life, The crucial question of how the separation or poses a lot of difficult questions. breaking away of the boundary layer from the The challenge is to derive quantitative results, boundary takes place is still not resolved. Oleı˘nik’s including estimates for the approximations. Ho- work concentrated on where and why separation mogenization, a special kind of averaging, started does NOT take place by proving that there exists as a mathematical discipline only about thirty years a unique boundary flow and hence one without ago, although in physics and engineering such av- separation, provided in our simple case that px =0 eraging methods had been used for many years to [O3]. In real life on an airplane wing this is determine effective properties and effective laws achieved by artificial suction, and Oleı˘nik exam- for heterogeneous media. ined this question in detail (see [O4]). In 1997, after The theory of homogenization is strongly linked many papers with others from 1963 on concern- to Oleı˘nik’s name. An impressive group of mathe- ing many different kinds of flow from magneto- maticians from the Department of Differential hydrodynamics to non-Newtonian fluids, she Equations at Moscow State University formed a wrote a book on the subject summarizing and leading team, and with them Oleı˘nik developed a proving many background results and her own key fruitful cooperation with France, Germany, and theorems [O5]. Italy, especially after the fall of the iron curtain. The long list of contributions of Oleı˘nik and her cowork- Singular Elliptic and Parabolic Theory5 ers, covering all of the main areas of homogeniza- Equations of elliptic or parabolic type enjoyed a tion, can be found in the monographs by Oleı˘nik, great deal of attention in the decades following Shamaev, and Yosifian [OSY] and by Jikov, Kozlov, World War II, and Oleı˘nik’s thesis was in this area. and Oleı˘nik [JKO]. These books are important She followed it with many other results, but she may sources of information and original ideas, and well be best remembered for her work on degen- cover important topics in theory and applications. erate problems where an elliptic equation becomes Oleı˘nik’s contributions were mainly in devel- parabolic at points or segments on the boundary oping the necessary tools to control the scale limit or even in whole patches of a domain. One of for initial and boundary value problems for systems the simplest examples is for the solutions u of with oscillatory coefficients or in domains with the wave equation depending only on x/t. Thus complex structures, holes, or oscillatory bound- with r = |x|, ρ = r/t, the equation for w = ru is aries. Essentially two methods are available: energy 2 (ρ − 1)wρρ + ρwρ − Λw =0, where Λ is the angu- methods, based on proper estimates of the solu- lar part of the Laplacian, and the degeneracy, not tions and compactness results; and multiscale surprisingly, occurs on the light cone ρ =1. expansions. Oleı˘nik determined the conditions for well- Introducing fast and slow (microscopic and posedness in many cases and in 1964 generalized macroscopic) variables, one starts from a formal and completed the problems posed by G. Fichera expansion with respect to the scale parameter, [F] (see [O6]). The technique was to add a term obtaining a recursive system for the coefficients

5CSM. 6WJ.

222 NOTICES OF THE AMS VOLUME 50, NUMBER 2 of the expansion. Oleı˘nik and her coworkers devel- In general, assumptions on the domain are nec- oped a systematic technique for determining essary: for example, the assumption that Ω is approximations and validating the expansion. In bounded and a Lipschitz domain. Also, an optimal the case of periodic structures, the approximation constant C is wanted, with the constant depending problem in the interior of the domain is reduced to on Ω. Kondratev and Oleı˘nik obtained the estimate solving a system with respect to fast variables in the by a rather simple proof, with asymptotically sharp scaled periodicity cell coupled with slow equations constants in the case of star-shaped domains. As in the whole domain. Effective equations can often Oleı˘nik liked to say, the original proof was nineteen be obtained by such averaging. The cited monographs pages, Friedrichs’s proof was nine pages, and they contain many examples. had reduced it to four pages [OSY, Chapter I, §2]. The analysis at the boundary or at interfaces is more complicated, and here the contributions of They also proved further Korn-type inequalities Oleı˘nik have been especially important. She and her that are even more useful. coworkers did pioneering work on the approximation References near the boundary. The analysis of boundary layers is a nontrivial problem in the case of periodic struc- [F] G. FICHERA, Sulle equazioni differenziali lineari ellittico- tures. Oleı˘nik and her coworkers obtained optimal paraboliche del secondo ordine, Atti Accad. Naz. Lin- results when the boundary is flat and in rational cei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I. (8) 5 (1956), 1–30. position with respect to the periodicity lattice. In [H] EBERHARD HOPF, The partial differential equation this situation an unbounded boundary-layer ut + uux = µuxx , Comm. Pure Appl. Math. 3 (1950), 201–30. cell has to be considered. Ideas developed in the [JKO] V. V. JIKOV, S. M. KOSLOV, and O. A. OLE˘ıNIK, Homog- papers of Oleı˘nik and Yosifian can be used to study enization of Differential Operators and Integral Func- problems in domains where the scale changes across tionals, translated from the Russian by G. A. Yosifian, an interface. Typical examples are processes in Springer-Verlag, 1994. partially porous or perforated domains where the [O1] O. A. OLE˘ıNIK, The place of the theory of differential derivation of effective transmission conditions equations in contemporary mathematics and its appli- and the estimate of the errors at the interface are cations (Russian), Differential Equations and Their Ap- the main aims. Flow and transport through filters plications, Moskov. Gos. Univ., Moscow, 1984, pp. 4–17. are practical, important examples. During her last [O2] ——— , Discontinuous solutions of non-linear differ- years, despite her serious health problems, Olga ential equations, Uspekhi Mat. Nauk 12 (1957), no. 3 Oleı˘nik was also involved with such multiscale (75), 3–73; English translation in Amer. Math. Soc. problems, strongly motivated by the many possible Transl. (2) 26 (1963), 95–172. applications. Oleı˘nik also developed a spectral [O3] ——— , On the equations of a boundary layer, Semi- theory adapted to homogenization. nari 1962/63 Anal. Algebra Geom. e Topol., Vol. 1, Ist. Stochastic homogenization, mainly in the case Naz. Alta Mat. Ediz. Cremonese, Rome, 1962/1963, of random coefficients, is the main topic in [JKO]. pp. 372–87. Several important contributions were obtained in [O4] ——— , On the mathematical theory of boundary Oleı˘nik’s group. Homogenization of stochastic layer for an unsteady flow of incompressible fluid, processes, also in random geometry, is a crucial Prikl. Mat. Mekh. 30, 801–21; English translation in topic of ongoing research, and here Oleı˘nik’s in- J. Appl. Math. Mech. 30 (1966), 951–74 (1967). fluence is clear. [O5] O. A. OLE˘ıNIK and V. N. SAMOKHIN, Mathematical Meth- ods in Boundary-Layer Theory, Fizmatlit “Nauka”, Problems in Elasticity7 Moscow, 1997; translated into English as Mathemati- Traditionally the links between mathematics and cal Models in Boundary Layer Theory, Chapman & mechanics are very strong in Russia, and Oleı˘nik’s Hall, 1999. mathematical research reflected this. Prominent [O6] O. A. OLE˘ıNIK, On a problem of G. Fichera, Dokl. Akad. Nauk SSSR 157 (1964), 1297–1300 (Russian). examples are Korn-type inequalities, basic in prov-  ing existence and estimating the solutions of [O7] O. A. OLE˘ıNIK and G. A. IOSIF JAN, Removable boundary the main boundary-value problems in elasticity. singularities and uniqueness of solutions of boundary Consider the tensor e(u) defined by value problems for second order elliptic and parabolic equations, Funktsional. Anal. i Prilozhen. 11 (1977), eij(u)=∂j ui + ∂iuj . no. 3, 54–67, 96 (Russian). [OSY] O. A. OLE˘ıNIK, A. S. SHAMAEV, and G. A. YOSIFIAN, Math- 2 Korn inequalities relate the L -norm of e(u) with ematical Problems in Elasticity and Homogenization, 2 the L -norm of ∇u, for instance in the form of the North-Holland, 1992. inequality [OV] O. A. OLE˘ıNIK and N. D. VVEDENSKAYA, The solution of Cauchy problem and boundary value problem for u H1(Ω) ≤ C( u L2(Ω) + e(u) L2(Ω)). nonlinear equations in the discontinuous functions class, Dokl. Akad. Nauk SSSR (N.S.) 113 (1957), 503–6 7WJ. (Russian).

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