Olga AlexandrovnaLadyzhenskaya(1922–2004)Susan Friedlander, Peter Lax, Cathleen Morawetz, Louis Nirenberg, Gregory Seregin, Nina Ural’tseva, and Mark Vishik

1320 NOTICES OF THE AMS VOLUME 51, NUMBER 11

Olga Alexandrovna La-dyzhenskaya died in hersleep on January 12th, 2004,in St. Petersburg, Russia, atthe age of eighty-one. She lefta wonderful legacy for math-ematics in terms of her fun-damental results connectedwith partial differential equa-tions and her “school” of stu-dents, collaborators, and col-leagues in Russia. In a lifededicated to mathematicsshe overcame personaltragedy arising from the cat-aclysmic events of twentiethcentury Russia to become oneof that country’s leadingmathematicians. Denied aplace as an undergraduate at

university, she was an exceptionally gifted younggirl, but one whose father disappeared in Stalin’sgulag. She eventually became a leading member ofthe Steklov Institute (POMI) and was elected to theRussian Academy of Science. Her mathematicalachievements were honored in many countries.She was a foreign member of numerous acade-mies, including the Leopoldina, the oldest Germanacademy. Among other offices, she was presidentof the Mathematical Society of St. Petersburg and,as such, a successor of Euler.

Ladyzhenskaya made deep and important con-tributions to the whole spectrum of partial differ-ential equations and worked on topics that ranged

from uniqueness of solutions of PDE to convergenceof Fourier series and finite difference approxima-tion of solutions. She used functional analytic tech-niques to treat nonlinear problems using Leray-Schauder degree theory and pioneered the theoryof attractors for dissipative equations. Develop-ing ideas of De Giorgi and Nash, Ladyzhenskaya andher coauthors gave the complete answer to Hilbert’snineteenth problem concerning the dependenceof the regularity of the solution on the regularityof the data for a large class of second-order ellip-tic and parabolic PDE. She published more than 250articles and authored or coauthored seven mono-graphs and textbooks. Her very influential book TheMathematical Theory of Viscous IncompressibleFlow, which was published in 1961, has become aclassic in the field. Her main mathematical “love”was the PDE of fluid dynamics, particularly theNavier-Stokes equation. This equation has a longand glorious history but remains extremely chal-lenging: for example, the issue of existence of phys-ically reasonable solutions to the Navier-Stokesequations in three dimensions was chosen as oneof the seven millenium “million dollar” prize prob-lems of the Clay Mathematical Institute. (The CMIwebsite gives a description by Fefferman of theprize problem.) The three-dimensional problemremains open to this day, although it was in the1950s that Ladyzhenskaya obtained the key resultof global unique solvability of the initial boundaryproblem for the two-dimensional Navier-Stokesequation. She continued to obtain influential resultsand to raise stimulating issues in fluid dynamics,even up to the days before her death.

DECEMBER 2004 NOTICES OF THE AMS 1321

This memorial article contains brief surveys ofsome of Ladyzhenskaya’s significant mathematicalachievements by two of her collaborators in St. Pe-tersburg, Gregory Seregin and Nina Ural’tseva.These are followed by “memories” of Olga Alexan-drovna by other distinguished mathematicians whohave known her and her work for many years,namely Peter Lax, Cathleen Morawetz, Louis Niren-berg, and Mark Vishik.

Olga Alexandrovna was a woman of great charmand beauty. We are very grateful to TamaraRozhkovskaya for presenting some of the lovelyphotographs of Olga that appear in the two volumesin honor of the eightieth birthday of Ladyzhenskayaedited by Rozhkovskaya, who is a “mathematicalgrandchild” of Olga and editor and publisher of theInternational Mathematical Series in which thesevolumes appear. At the conclusion of her appreci-ation article in the second of these two volumes,Nina Ural’tseva ( a “mathematical child” and closefriend and collaborator of Olga ) writes “In the Sci-ence Museum in Boston there is an exhibition de-voted to mathematics. The names of the most in-fluential mathematicians of the 20th century arecarved on a large marble desk…and Olga La-dyzhenskaya is among them”.

—Susan Friedlander

Gregory Seregin and Nina Ural’tseva

Olga Alexandrovna Ladyzhenskaya passed awayquite unexpectedly on January 12 this year. She wasin good spirits two days before that. She had justsketched a paper on some computational aspectsin hydrodynamics and planned to finish it inFlorida. Olga was fond of sunlight; therefore, everywinter, the darkest period in our St. Petersburg, shelonged for the South…. In her last years she hadserious problems with her eyes, and she sufferedenormously because of the winter darkness. She re-sisted such problems by all means, for example,using special pencils when writing. She was to leavefor Florida on January 12. On the eve of January11 she went to rest before her long trip and did notwake up. God gave her an easy death.

Olga Ladyzhenskaya was a brilliant mathemati-cian and an outstanding person who was admiredby distinguished scientists, writers, artists, andmusicians, often becoming a source of inspiration

for them. The eminent mathematician V. Smirnov;the great pianist and professor of the Academy ofMusic, N. Golubovskaja; the distinguished world-famous poets A. Akhmatova and J. Brodsky; andthe famous writer A. Solzhenitsyn are found amongthem.

Her influence on twentieth century mathemat-ics is highly valued by the world scientific com-munity. It was not only Olga’s scientific results,though truly deep and fundamental, but also herpersonal integrity and energy that played an es-pecial role in her contribution to mathematics.

Olga’s focus in life was not limited to mathe-matics and science. She was deeply interested inarts and intellectual life in general. She loved ani-mals, was a passionate mushroom lover, and en-joyed flowers. She was an enthusiastic traveler andhad the wonderful skill of a storyteller when shar-ing her impressions with friends.

There were few things that did not touch her;she reacted keenly to any injustice, to the misfor-tunes of others; and she helped lone and feeble peo-ple. She took this very personally.

She expressed openly her views on social mat-ters, even in the years of the totalitarian politicalregime, often neglecting her own safety.

Olga grew up during very hard times in Russia.She was born on March 7, 1922, in the tiny townof Kologriv in the north part of Russia. Her father,Alexander Ivanovich, taught mathematics in a highschool. Her mother, Anna Mikhailovna, kept house

Selected Honors of Olga Ladyzhenskaya1969 The State Prize of the USSR1985 Elected a foreign member of the Deutsche Akademie

Leopoldina1989 Elected a member of the Accademia Nazionale dei Lincei1990 Elected a full member of the Russian Academy of Science2002 Awarded the Great Gold Lomonosov Medal of the Russian

Academy2002 Doctoris Honoris Causa, University of Bonn

Susan Friedlander, is professor of mathematics at the Uni-versity of Ill inois-Chicago. Her email address issusan@math.uic.edu.

Gregory Seregin is professor of mathematics at the St. Pe-tersburg Department of the Steklov Institute. His email ad-dress is seregin@pdmi.ras.ru.

Nina Ural’tseva is professor of mathematics at St. Peters-burg University, Russia. Her email address is uraltsev@euclid.pdmi.ras.ru.

Left to right, Nina Ural’tseva, Olga Ladyzhenskaya, V. Smirnov.

1322 NOTICES OF THE AMS VOLUME 51, NUMBER 11

and looked after her husbandand three daughters. Olga’s“grandfather”1, Gennady La-dyzhensky, was a famouspainter. There were manybooks, including books onhistory and fine arts, in theirhouse. Books were almost theonly source of cultural edu-cation, because Kologriv wastoo far from cultural centers.The town was situatedamong wild forests, near thepicturesque river Unzha. Allher life Olga carefully keptbeautiful landscape paintingsby her grandfather, some ofthem depicting fine views ofthe Unzha.

It was in the summer of 1930 that AlexanderIvanovich decided to teach mathematics to his owndaughters. After he madesome explanation of thebasic notions of geome-try, he formulated a theo-rem and suggested thathis daughters prove itthemselves. It turned outthat the youngest one,Olga, was the best of hisstudents. She loved to dis-cuss mathematics withher father, and soon theystudied calculus together.In October 1937 Alexander Ivanovich was arrestedand soon killed by NKVD, the forerunner of the KGB.It was a great shock for the family. Later, in 1956,he was officially exonerated “due to the absence ofa corpus delicti”. His death put the family in a verydistressed situation. The mother and older daugh-ters had to go to work. Olga’s mother had grownup in a small village in Estonia and could do craftwork. She made dresses, shoes, soap, and manyother things. That was the way the family survivedin those days.

In 1939 Olga graduated with honors from Kolo-griv High School and went to Leningrad to continueher education. However, it was forbidden for a per-son whose father was considered an “enemy of hisNation” to enter Leningrad State University.2 Shewas accepted at Pokrovskii Pedagogical Institute,and finished two years of studies there in June1941. The war forced her to leave Leningrad. In the

fall of 1941 she worked as a teacher in the townof Gorodets, and in the spring of 1942 she re-turned to Kologriv, where she took her father’sposition as a high school mathematics teacher. InOctober 1943 she finally became a student atMoscow State University and graduated in 1947.During these years, I. G. Petrovskii was her adviser.At that time she was strongly influenced byGelfand’s seminar.

In 1947 Olga married A. A. Kiselev, a Leningradresident, so she moved there, with a recommen-dation from Moscow State University to the grad-uate school of Leningrad State University (LGU). AtLGU S. L. Sobolev was appointed to be her scien-tific adviser. From the time she came to LGU, a veryclose friendship developed between Olga and V. I.Smirnov and his family. In the fall of 1947 Smirnov,at Ladyzhenskaya’s request, became head of a sem-inar on mathematical physics, which has been ac-tive ever since. The seminar brought together manymathematicians of the city working in the area ofpartial differential equations and their applica-tions. Until her last days Ladyzhenskaya was oneof the leading participants.

Finite-Difference Method and HyperbolicEquationsEven the first results that Ladyzhenskaya obtainedin the late 1940s and in the early 1950s were abreakthrough in the theory of PDEs. In her Ph.D.thesis, defended at LGU in 1949, she proposed adifference analog of Fourier expansions for peri-odic functions on grids and investigated their con-vergence as the grid step goes to zero in the dif-ference analogues of the space Wk

2 . Using theseexpansions, she investi-gated various differenceschemes for model equa-tions, distinguished thosethat are stable for certainratios of grid steps in thespace and time variables,and then proved the sta-bility of these schemes forequations with variablecoefficients. In particular,she gave a simpler proofof unique local solvabilityof the Cauchy problem forhyperbolic quasilinear sys-

tems than the proof that Petrovskii had given.After defending her Ph.D. thesis, Ladyzhenskaya

decided to study initial boundary-value problemsfor linear hyperbolic equations of second order. Shestarted with justification of the Fourier method. Inher publications of 1950–1952, Ladyzhenskayagave exhaustive answers concerning series expan-sions of functions in Wk

2 (Ω) in the eigenfunctionsof arbitrary symmetric second-order elliptic

1Olga’s actual grandfather was Ivan whose brother wasthe painter Gennady Ladyzhensky. Gennady lived with Ivanand his family. Olga and her sisters always called Gennady“Dedushka” which means grandfather.2She had passed the entrance exams to this university whichat the time was considered the best in the Soviet Union.

Olga as a baby, with her father,mother, and two sisters.

DECEMBER 2004 NOTICES OF THE AMS 1323

operators in a bounded domain Ω ∈ Rn , under anyof the classical boundary conditions on ∂Ω. More-over, in [1] she found a solution to the problem ofdescribing the domain of the closure in L2(Ω) ofan elliptic operator L with the Dirichlet boundarycondition. The solution is based on the inequality

(1) ‖u‖W 22 (Ω) ≤ C(Ω)(‖Lu‖L2(Ω) + ‖u‖L2(Ω)),

proved by Ladyzhenskaya. Here, L is a general sec-ond-order elliptic operator with smooth coeffi-cients in front of the second derivatives, Ω is abounded domain in Rn with smooth boundary,and u is an arbitrary function in W2

2 (Ω) that van-ishes on the boundary or satisfies a nondegener-ate homogeneous boundary condition of the firstorder. The significance of this result for the the-ory of differential operators, including spectraltheory, can hardly be overestimated.

Most of the results mentioned above were in-cluded in her first monograph [2], published in1953. Unfortunately, this book was not translatedinto European languages.

Thanks to this work, the notion of a weak solu-tion to an initial boundary-value problem becomesan important concept in mathematical physics.Systematic investigation of the entire scale of weaksolutions in various function spaces, masterly an-alytic techniques for obtaining estimates for the in-tegral norms, together with general arguments offunctional analysis, provided Ladyzhenskaya witha strong background that led to success in thestudy of the solvability of the boundary-value prob-lems and initial boundary-value problems for lin-ear PDEs of classical type.

References[1] O. A. LADYZHENSKAYA, On some closure of an elliptic op-

erator, Dokl. Akad. Nauk SSSR 79 (1951), 723–725.(Russian)

[2] ——— , The Mixed Problem for a Hyperbolic Equation,Gostehteorizdat, Moscow, 1953.

Quasilinear PDEs of Elliptic and ParabolicTypesAn extensive series of joint papers by Ladyzhen-skaya and Nina Ural’tseva was devoted to the in-vestigation of quasilinear elliptic and parabolicequations of the second order. At the beginning ofthe last century S. N. Bernstein proposed an ap-proach to the study of the classical solvability ofboundary-value problems for such equations basedon a priori estimates for solutions. He also de-scribed conditions that are in a certain sense nec-essary for such solvability. These conditions, whichare usually called natural growth restrictions, areformulated in terms of growth orders of functionsentering the equations with respect to gradients ofsolutions. Due to works by Leray and Shauder, in-vestigation of the classical solvability of the

Dirichlet problem was reduced to obtaining a pri-ori bounds for its solutions in C1+α norm. Up tothe mid-1950s the program had been realized forthe Dirichlet problem for two-dimensional ellipticequations, but even in those cases some unnaturalconditions were supposed.

The conception of a generalized solution firstshowed up in the calculus of variations. The effortsof many outstanding mathematicians such asHilbert, Tonelli, Morrey, and A. G. Sigalov led to theconstruction of direct methods for proving the ex-istence of solutions to the Euler equations of thecorresponding variational integrals. In contrast tothe existence of minimizers, the number of inde-pendent variables plays a significant role in thestudy of their differentiability properties. By the be-ginning of the 1940s Morrey was investigating theregularity for the case n = 2.

The attack on higher-dimensional problemsstarted in the mid-1950s.In 1955, at the Depart-ment of Physics of LGU,Ladyzhenskaya deliv-ered a special course oflectures about her newresults on quasilinearparabolic equations,later published in [1] and[2]. These papers con-tained a rather generalform of the method ofauxiliary functions,which goes back to Bern-stein’s work. Ladyzhen-skaya set herself the taskof unraveling the secretof finding auxiliary func-tions for the evaluation

of the maximum modulus of the gradient of the so-lution in the multidimensional case. She reducedit to solving a nonlinear differential inequality forfunctions of one variable and proved its solvabil-ity under a certain condition of smallness. Thismade it possible to estimate the spatial gradientfor any solution in terms of its modulus of conti-nuity under natural growth conditions. In papers[1] and [2], she applied her result to a narrower classof quasilinear elliptic and parabolic equations thatdoes not require a preliminary estimate for the os-cillations of the solution. Almost simultaneouslywith [1], in papers by J. Nash and E. de Giorgi, abound for the Hölder norm of solutions was es-tablished for the simplest case of linear parabolicand elliptic equations having a divergence form withbounded measurable coefficients.

De Giorgi’s ideas were further developed in[3]–[8]. The results of these investigations were themain content of monographs [9]–[11] on the the-ory of quasilinear equations of elliptic and

1324 NOTICES OF THE AMS VOLUME 51, NUMBER 11

parabolic types (the latter written jointly with Solon-nikov). The books presented a rather complete the-ory for quasilinear equations of divergence formand, in particular, global solvability of the classi-cal boundary-value problems under natural growthrestrictions. Moreover, the methods developed bythe authors in [9] enabled them to analyze the de-pendence of smoothness of generalized solutionson the smoothness of data for equations of diver-gence form. It was proved that under natural growthrestrictions, the regularity of weak extremals forthe higher-dimensional variational is completely de-termined by the smoothness of the integrand. In asense this concluded the investigation of Hilbert’snineteenth and twentieth problems for second-order equations. Also, classes of quasilinear sys-tems with diagonal principal part and a specialstructure of the terms quadratic in gradients weresingled out there, and it was shown that they shareproperties that are characteristic of scalar diver-gence equations. Such systems later became the ob-ject of detailed studies in research on harmonicmaps of manifolds.

The books [9]–[11] also presented a number ofdeep results in the theory of quasilinear equationsof general nondivergence form. However, for suchequations it was only later, at the beginning of the1980s, that unnatural restrictions were completelyremoved and the results brought to the same levelof generality as in the divergence form case. Thisrequired techniques developed by N. V. Krylov andM. V. Safonov for investigating linear equations ofnondivergence form with bounded measurable co-efficients. In [12]–[14], the conditions on the datawere made even weaker, the presence of integrablesingularities with respect to the independent vari-ables was allowed, and the solvability of the Dirich-let problem was studied in Sobolev spaces.

The methods developed in the monographs[9]–[11] turned out to be effective also in the studyof broader classes of equations that contain non-uniformly elliptic operators like mean curvature op-erator [15]. The joint work of Ladyzhenskaya withN. M. Ivochkina was devoted to geometric topics aswell. In 1994–1997, they published a series of pa-pers investigating the global classical solvability ofthe first initial boundary-value problem for non-linear equations of parabolic type that describethe flows generated by the symmetric functions ofthe Hessian of the unknown surface or by its prin-cipal curvatures.

References[1] O. A. LADYZHENSKAYA, The first boundary value problem

for quasi-linear parabolic equations, Dokl. Akad. NaukSSSR 107 (1956), 636–638. (Russian)

[2] ——— , Solution of the first boundary value problemin the large for quasi-linear equations, Trudy Moskov.Mat. Obshch. 7 (1958), 219–220. (Russian)

[3] O. A. LADYZHENSKAYA and N. N. URAL’TSEVA, A variationalproblem and quasi-linear elliptic equations in manyindependent variables, Dokl. Akad. Nauk SSSR 135(1960), 1330–1333; English transl., Soviet Math. Dokl.1 (1960), 1390–1394.

[4] ——— , Quasi-linear elliptic equations and variationalproblems in several independent variables, UspekhiMat. Nauk 16:1 (1961), 19–90; English transl., RussianMath. Surveys 16:1 (1961), 17–91.

[5] ——— , A boundary value problem for linear and quasi-linear parabolic equations, Dokl. Akad. Nauk SSSR139 (1961), 544–547; English transl., Soviet Math. Dokl.2 (1961), 969–972.

[6] ——— , On the smoothness of weak solutions of quasi-linear equations in several variables and of variationalproblems, Comm. Pure Appl. Math. 14 (1961), 481–495.

[7] ——— , Boundary value problems for linear and quasi-linear parabolic equations. I, Izv. Akad. Nauk SSSRSer. Mat. 26 (1962), 5–52; English transl., Amer. Math.Soc. Transl. (2) 47 (1966), 217–267.

[8] ——— , Boundary value problems for linear and quasi-linear parabolic equations. II, Izv. Akad. Nauk SSSR Ser.Mat. 26 (1962), 753–780; English transl., Amer. Math.Soc. Transl. (2) 47 (1966), 268–299.

[9] ——— , Linear and Quasi-Linear Elliptic Equations,Nauka, Moscow, 1964; English transl., Academic Press,New York, 1968.

[10] ——— , Linear and Quasi-Linear Elliptic Equations,2nd rev. ed., Nauka, Moscow, 1973. (Russian)

[11] O. A. LADYZHENSKAYA, V. A. SOLONNIKOV, and N. N.URAL’TSEVA, Linear and Quasi-Linear Equations of Par-abolic Type, Nauka, Moscow, 1967; English transl.,Amer. Math. Soc., Providence, RI, 1968.

[12] O. A. LADYZHENSKAYA and N. N. URAL’TSEVA, Solvabilityof the first boundary value problem for quasi-linearelliptic and parabolic equations in the presence ofsingularities, Dokl. Akad. Nauk SSSR 281 (1985),275–279; English transl., Soviet Math. Dokl. 31 (1985),296–300.

[13] ——— , A survey of results on the solvability of bound-ary value problems for uniformly elliptic and parabolicsecond-order quasi-linear equations having unboundedsingularities, Uspekhi Mat. Nauk 41:5 (1986), 59–84;English transl., Russian Math. Surveys 41:5 (1986),1–31.

[14] ——— , Estimates on the boundary of the domain forthe first derivatives of functions satisfying an ellipticor a parabolic inequality, Trudy Math. Inst. Steklov. 179(1988), 102–125; English transl., Proc. Steklov Inst.Math. 1989, no. 2, 109–135.

[15] ——— , Local estimates for gradients of solutions ofnon-uniformly elliptic and parabolic equations, Comm.Pure Appl. Math. 23 (1970), 677–703.

HydrodynamicsThe mathematical theory of viscous incompress-ible fluids was the favorite topic of Olga La-dyzhenskaya. She was involved with this activityfrom the middle of the 1950s till the very end ofher life. Her personal contributions to mathemat-ical hydrodynamics were deep and very impor-tant. But what was perhaps even more importantis that she was a great source of new problems forothers.

DECEMBER 2004 NOTICES OF THE AMS 1325

In the middle of the 1950s, Ladyzhenskaya beganwith the simplest case, the stationary Stokes sys-tem:

(0.1)µ∆u−∇p = −f

div u = 0

in Ω, u|∂Ω = 0

Here, Ω is a domain in Rn(n = 2,3), f ∈ L2(Ω; Rn)is a given force, µ is constant viscosity, u and p areunknown velocity field and pressure. If we defineby H(Ω) and H1(Ω) the closures of the set

C∞0 (Ω; Rn) = v ∈ C∞0 (Ω; Rn) ‖ div v = 0

in L2(Ω; Rn) and W12 (Ω; Rn) , respectively, the ve-

locity can be determined from the variational iden-tity

(0.2) µ∫Ω

∇u : ∇v dx =∫Ω

f ·v dx, ∀v ∈ H1(Ω).

At that time, Ladyzhenskaya had already been ableto prove that the solution u to (0.2) has the secondderivatives which are locally square summable inΩ. In order to recover the pressure, she proved thefollowing fundamental fact:

Theorem 0.1. Let w ∈ L2(Ω; Rn) be orthogonal toall v ∈

C∞0 (Ω; Rn) , then w = ∇p for some

p ∈ L2,loc(Ω) with ∇p ∈ L2(Ω; Rn).

The theorem was proved in the celebrated mono-graph [1], which contains the main results obtainedby Ladyzhenskaya in the 1950s on Stokes andNavier-Stokes equations.

In 1957, the joint paper [2] by Ladyzhenskayaand Kiselev was published, in which they consid-ered the first initial boundary value problem for theNavier-Stokes equations:

(0.3)∂tv + v · ∇v − µ∆v +∇p = f

div v = 0

in QT = Ω×]0, T [

v|∂Ω×[0,T ] = 0, v|t=0 = a

and proved unique solvability of it in an arbitrarythree-dimensional domain. At that time, the onlyone result of such kind was the pioneering result[3] of J. Leray on the Cauchy problem. All statementsin [2] were proved without any special representa-tion of solutions but allowed freedom in the choicefor a class of generalized solutions in which theuniqueness theorem is preserved. In particular,they worked in the class W1

2 (QT )∩L∞(0, T ;L4(Ω)) for v , with ∂t∇v ∈L2(QT ) , andshowed that unique solvability in the class on a non-empty interval ]0, T [ , where T depends on W2

2 -normof a and on L2-norms of the known functions f and∂tf . If these norms are sufficiently small, thenT = +∞ . In this paper, they cited Hopf’s work [5],

saying that the class of weak solutions introducedby E. Hopf is too wide to prove uniqueness.

One of the most remarkable results, proved byLadyzhenskaya at the end of the 1950s and pub-lished in [4], was the global unique solvability ofthe initial boundary problem for the 2D Navier-Stokes equations. For the Cauchy problem, it wasestablished by J. Leray. The proof was based onideas developed in [2] and on new types of in-equalities, which nowadays are called multiplica-tive inequalities. In particular, Ladyzhenskayaproved that, for any v ∈ C∞0 (Ω), it holds:

(0.4) ‖v‖2L4(Ω) ≤ C‖∇v‖L2(Ω)‖v‖L2(Ω).

Here, Ω is an arbitrary domain in R2 and C is a uni-versal constant. The inequality (0.4) now bears hername. At almost the same time, Lions and Prodiproved in [6] the global unique solvability of thetwo-dimensional problem in a different way butwith the help of Ladyzhenskaya’s inequality (0.4).

As to the three-dimensional case, Ladyzhen-skaya did not pay special attention to additionalconditions that provide uniqueness of the so-calledweak Leray-Hopf solutions. A function u is calleda weak Leray-Hopf solution if it has the finite en-ergy, satisfies a certain variational identity in whichtest functions are divergence free and compactlysupported in the space-time cylinder QT, is con-tinuous in time with values in L2(Ω) equipped withthe weak topology, and satisfies the energy in-equality for all moments of time and the initial con-dition in L2-sense. It was mentioned in the first edi-tion of her monograph [1] that the additionalcondition v ∈ L4,8(QT ) ≡ L8(0, T ;L4(Ω)) givesuniqueness in the class of weak Leray-Hopf solu-tions. This condition is a particular case of the fol-lowing theorem proved essentially by Prodi in [7]and by Serrin in [8].

Ladyzhenskaya near the River Dnieper in theUkraine, while on a break from a mathconference.

1326 NOTICES OF THE AMS VOLUME 51, NUMBER 11

Theorem 0.2. Assume that v is a weak Leray-Hopfsolution and

(0.5) v ∈ Ls,l(QT )

for positive numbers s and l satisfying the condi-tion

(0.6)3s

+2l≤ 1, s > 3.

Then any other weak Leray-Hopf solution to the sameinitial boundary-value problem coincides with v .

Ladyzhenskaya later included results of this typein the second Russian edition [9] of her mono-graph.

In 1967, Ladyzhenskaya proved in [10] that infact weak Leray-Hopf solutions satisfying condi-tions (0.5) and (0.6) are smooth.

Theorem 0.3. Assume that all conditions of Theo-rem 0.2 are fulfilled. Let, in addition, f ∈ L2(QT ) ,a ∈ H1(Ω) , and Ω is the bounded domain in R3ofclass C2. Then ∂tv, ∇2v, and ∇p are in L2(QT ) .

As it was indicated in one of the last papers by La-dyzhenskaya [11], this theorem might be a goodguideline for those who would like to solve the mainproblem of mathematical hydrodynamics. By theway, Ladyzhenskaya formulated it as follows: Oneshould look for the right functional class in whichglobal unique solvability for initial boundary valueproblems for the Navier-Stokes takes place.

Up to now, very few qualitatively new results onglobal unique solvability results for the three-dimensional nonstationary problem are known.The most impressive among them is Ladyzhen-skaya’s result for the Cauchy problem in the classof radially asymmetric flows with zero angular ve-locity. This was proved in [12].

As we mentioned above, Ladyzhenskaya be-lieved that in the 3D case the Navier-Stokes equa-tions, even with very smooth initial data, do not pro-vide uniqueness of their solutions on an arbitrarytime interval. Hopf and later other mathematicians,Ladyzhenskaya among them, tried unsuccessfullyto state a reasonable principle of choice for deter-minacy. Giving up these attempts, Ladyzhenskayaintroduced the so-called modified Navier-Stokesequations, which are now known as Ladyzhen-skaya’s model. These equations differ from theclassical ones only for large-velocity gradients. Forthem, she proved global unique solvability underreasonably wide assumptions on the data of theproblem. These results were the main content ofher report to the International Congress of Math-ematicians in 1966. For a long time, problems ofthis type were not very popular due to their diffi-culties, and it is only in the last decade that theyhave attracted the attention of many mathemati-cians trying to improve on Ladyzhenskaya’s old re-sults.

At the end of this section, we would like to men-tion the remarkable results Ladyzhenskaya ob-tained on the solvability of stationary problems forthe Navier-Stokes equations. The first attempts inthis direction were made by Odqvist and Leray.However, the question of global solvability (for allvalues of Reynolds numbers) had remained opentill the end of the 1950s. Ladyzhenskaya’s ap-proach was based on her conception of weak so-lution and a good choice of the energy space. Thisturned out to be the space H1(Ω) , the closure ofC∞0 (Ω) in the Dirichlet integral norm. Using knowna priori estimates for the velocity gradient in L2(Ω),she proved in [13] and [14] the existence of at leastone solution in the energy space H1(Ω) . Later, inthe 1970s, Ladyzhenskaya, together with Solon-nikov, successfully studied stationary problems indomains with noncompact boundaries.

References[1] O. LADYZHENSKAYA, The Mathematical Theory of Viscous

Incompressible Flow, Fizmatgiz, 1961; English transl.,Gordon and Breach, New York, 1969.

[2] O. A. LADYZHENSKAYA and A. A. KISELEV, On the existenceand uniqueness of the solution of the non-stationaryproblem for a viscous incompressible fluid, Izv. Akad.Nauk SSSR Ser. Mat. 21 (1957), 665–680; English transl.,Amer. Math. Soc. Transl. (2) 24 (1963), 79–106.

[3] J. LERAY, Sur le mouvement d’un liquide visqueux em-plissant l’espace, Acta Math. 63 (1934), pp. 193–248.

[4] O. A. LADYZHENSKAYA, Solution “in the large” of bound-ary value problems for the Navier-Stokes equations intwo space variables, Dokl. Akad. Nauk SSSR 123 (1958),427–429; English transl., Soviet Phys. Dokl. 3 (1959),1128–1131 and Comm. Pure App. Math. 12 (1959),427–433.

[5] E. HOPF, Über die Anfangswertaufgabe für die hydro-dynamischen Grundgleichungen, Math. Nachrichten 4(1950–51), 213–231.

Ladyzhenskaya, age 79, in her St. Petersburg apartment.

DECEMBER 2004 NOTICES OF THE AMS 1327

[6] J.-L. LIONS and G. PRODI, Un théorème d’existence et unic-ité dans les équations de Navier-Stokes en dimension2, C. R. Acad. Sci. Paris 248 (1959), 3519–3521.

[7] G. PRODI, Un teorema di unicità per el equazioni diNavier-Stokes, Ann. Mat. Pura Appl. 48 (1959), pp.173–182.

[8] J. SERRIN, The initial value problem for the Navier-Stokes equations, Nonlinear Problems (R. Langer, ed.),pp. 69–98, The University of Wisconsin Press, Madi-son, 1963.

[9] O. A. LADYZHENSKAYA, Mathematical Problems of the Dy-namics of Viscous Incompressible Fluids (2nd edition)Nauka, Moscow, 1970.

[10] ——— , Uniqueness and smoothness of generalizedsolutions of the Navier-Stokes equations, Zap. Nauchn.Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 5(1967), 169–185; English transl., Sem. Math. V. A.Steklov Math. Inst. Leningrad 5 (1969), 60–66.

[11] ——— , The sixth problem of the millenium: TheNavier-Stokes equations, existence and smoothness,Uspekhi Mat. Nauk 58:2(2003), 181–206; English transl.,Russian Math. Surveys 58:2(2003), 395–425.

[12] ——— , Unique global solvability of the three-dimen-sional Cauchy problem for the Navier-Stokes equationsin the presence of axial symmetry, Zap. Nauchn. Sem.Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 7 (1968),155–177; English transl., Sem. Math. V. A. Steklov Math.Inst. Leningrad 7 (1970), 70–79.

[13] ——— , A stationary boundary value problem for vis-cous incompressible fluid, Uspekhi Mat. Nauk, 13(1958), 219–220 (Russian).

[14] ——— , Stationary motion of viscous incompressiblefluids in pipes, Dokl. Akad. Nauk SSSR 124 (1959),551–553; English transl., Soviet Physics Dokl. 124(4)(1959), 68–70.

Attractors for PDEsIn [1], Ladyzhenskaya considered a semigroup gen-erated by the two-dimensional initial boundary-value problem for the Navier-Stokes equations.Solution operators were defined as Vt (a) = v(t), andthe phase space was H(Ω). It was assumed thatthe force f in (0.3) is independent of time and be-longs to L2(Ω). It was known that solution opera-tors are continuous in (a, t) on H(Ω)×R1

+ and eachbounded set B ⊂ H(Ω) is pulled into the ballB(R0) =u ‖ ‖u‖L2(Ω) < R0 ⊂ H(Ω) of radius R0 >(λµ)−1 ‖f‖L2(Ω) in a finite time, where λ is the firsteigenvalue of the Stokes operator for the domainΩ, and B stays there for all large t . She showed thatsolution operators are compact on H(Ω) for eachfixed positive t. For finite-dimensional phase spacesthis result is immediate, but in the case of infinite-dimensional phase spaces that was a very impor-tant step. Then Ladyzhenskaya took the intersec-tion

(0.1) M =⋂t>0

Vt (B(R0))

and proved that the set M is nonempty and com-pact and attracts any bounded subsets of the phasespace. She found many other interesting proper-

ties of M. It is invariant, each trajectory on M canbe extended to negative times, and in a sense it isfinite-dimensional. The latter means that there isa number N with the following property: If for twofull trajectories lying in M theprojections on the finite dimen-sional space spanned by the firstN eigenfunctions of the Stokes op-erator coincide, then the trajecto-ries themselves coincide. In theconclusion of this remarkablepaper, it was claimed that the pro-posed approach is applicable tomany other dissipation problems,in particular, to problems of par-abolic types. So, [1] opened a chap-ter in the theory of PDE, namelythe theory of stability “in thelarge”. This program was realizedin her last monograph [3].

In her survey article [2], sheproposed to call the set M a “global minimal B -attractor”. Her explanation of that was as follows.The word “minimal” indicates that no proper sub-set of M attracts the whole of H(Ω), and the let-ter B emphasizes that any bounded set in H(Ω) isuniformly attracted to M. Certainly, this makessense.

References[1] O. A. LADYZHENSKAYA, A dynamical system generated by

the Navier-Stokes equations, Zap. Nauchn. Sem.Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 27 (1972),91–115; English transl., J. Soviet. Math. 3 (1975),458–479.

[2] ——— , Finding minimal global attractors for the Navier-Stokes equations and other partial differential equa-tions, Uspekhi Mat. Nauk 42:6, 1987, 25–60; Englishtransl., Russian Math. Surveys 42:6, 1987, 27–73.

[3] ——— , Attractors for semigroups and evolution equa-tions, Lezioni Lincei, 1988, Cambridge Univ. Press,Cambridge, 1991.

M. I. Vishik

I first met Olga Alexandrovna Ladyzhenskaya in1945, and we became friends at once. She was thena third-year student at Moscow State University. Weoften met at the seminars, and we discussed math-ematical problems while walking along the widecorridors of the Department of Mechanics andMathematics of the university.

Her scientific adviser at Moscow State Universitywas Ivan Georgievich Petrovskiı . In 1946 Israel

Mark Vishik is professor of mathematics at the MoscowState University and at the Institute of Information Trans-mission Problems of the Russian Academy of Science. Hisemail address is vishik@iitp.ru.

1328 NOTICES OF THE AMS VOLUME 51, NUMBER 11

Moiseevich Gelfand organized a seminar for threeyoung mathematicians: O. A. Ladyzhenskaya, O. A.Oleınik, and myself. As a result each participant ofthe seminar got an interesting mathematical prob-lem to seriously work on. O. A. Ladyzhenskaya gotthe problem of describing the domain of an ellip-tic operator of the second order with Dirichletboundary conditions and with right-hand side be-longing to L2(Ω). She proved that if this boundary-value problem has a unique solution, then the el-liptic operator is an isomorphism between theSobolev space H2(Ω)∩H1

0 (Ω) and L2(Ω). She alsofound an estimate for the norm for the corre-sponding inverse operator.

In the late forties Olga Alexandrovna moved toLeningrad, where she became a postgraduate stu-dent of Vladimir Ivanovich Smirnov. VladimirIvanovich admired her talent, and she took an ac-tive part in his seminar. Later Olga Alexandrovnabecame the head of the seminar. From time to timeOlga Alexandrovna organized conferences on dif-ferential equations and their applications. Theseconferences were very popular. Many mathemati-

cians from other universi-ties and institutes in the So-viet Union used to take partin them.

In Leningrad OlgaAlexandrovna married An-drey Alekseevich Kiselev,who looked very much likePierre Bezukhov, the heroof the novel War and Peaceby Leo Tolstoy. My wife andI often visited them whenwe were in Leningrad. Wewitnessed how tender andloving Andrey Alexeevichwas. She literally meant theworld to him. However, theyseparated later. AndreyAlexeevich wanted to havechildren, but Olga Alexan-

drovna did not. She justified her decision by herwish to devote her life to mathematics only, towhich children might be an obstacle. Thus, OlgaAlexandrovna stayed single for the rest of her life.

For many years starting in the late forties, OlgaAlexandrovna visited us many times, and some-times even stayed with us. She was usually inter-ested in what new mathematical ideas I had workedout over the summer, and she told me, in turn, ofher own new achievements. We also used to discusswith her new compelling and timely mathematicalproblems; the connections between functionalanalysis and the theory of differential equations andmany other things. Ladyzhenskaya and I publisheda paper on these problems in Uspekhi Matem-aticheskih Nauk (see [1]).

In 1953, at Moscow State University, Olga Alexan-drovna defended her “habilitation” dissertationdevoted to the problem of regularity of solutionsof a mixed boundary-value problem for a generalhyperbolic equation of the second order. She foundrather sharp conditions guaranteeing that the so-lutions of these equations are the solutions in theclassical sense. Olga Alexandrovna justified theFourier method for the solution of hyperbolicboundary-value problems. Furthermore, she also

studied the applicationof the method of theLaplace transform tothese equations. Allthese problems were ex-pounded in her book [2].

In the fifties and six-ties Olga Alexandrovnaoften gave talks at theseminars of IvanGeorgievich Petrovskiıat Moscow State Uni-versity. I recall howgreatly the listenerswere impressed by her

proof of the uniqueness theorem for the initialboundary-value problem for the two-dimensionalNavier-Stokes system. Since the existence theoremfor these equations was proved earlier, the well-posedness of the main boundary-value problemfor the Navier-Stokes system in two dimensions wasestablished thanks to the remarkable result of OlgaAlexandrovna. The uniqueness theorem for thethree-dimensional Navier-Stokes system is still notproved. Olga Alexandrovna devoted her book onfluid dynamics to three people she most respected,namely, her father, Vladimir Ivanovich Smirnov,and Jean Leray. In this book she studied in greatdetail many problems related to the stationary andnonstationary Navier-Stokes system.

In collaboration with Nina Nikolaevna Ural’tseva,Olga Alexandrovna wrote a book [4] on linear andquasilinear elliptic equations. Many fundamentalresults were obtained in this book, which remainsa real encyclopedia on the subject. In the sixties,Olga Alexandrovna, Nina Nikolaevna Uralt’seva,and Vsevolod Alekseevich Solonnikov wrote a book[5] on linear and quasilinear parabolic equations.In section 2 of this memorial article, Nina Niko-laevna writes more about this wonderful mono-graph. Olga Alexandrovna is also the author of thebook [6] on the attractors of semigroups and evo-lution equations. This book is based on a lecturecourse that Olga Alexandrovna gave in different uni-versities in Italy, the so-called Lezioni Lincei.

Olga Alexandrovna was a very educated andhighly cultured person. The famous Russian poet,Anna Akhmatova, who knew Ladyzhenskaya verywell, devoted a poem to her.

Olga Ladyzhenskaya with herhusband, Andrey Kiselev.

DECEMBER 2004 NOTICES OF THE AMS 1329

Olga Alexandrovna was once a member of thecity council of people’s deputies. She helped manymathematicians in Leningrad to obtain apartments(free of charge) for their families.

Once in the Steklov Mathematical Institute inLeningrad, removing my coat in the cloakroom, Isaid jokingly to the cloakroom attendant that Iwas going to take away Olga Alexandrovna toMoscow with me, and I got a serious answer: “Wewill never give up our Olga Alexandrovna!”

Olga Alexandrovna was a deeply religiouswoman.

Olga Alexandrovna devoted her life to mathe-matics, sacrificing her own happiness for it. Now,we can surely say that she has become a “classicfigure” of mathematical science.

References[1] M. I. VISHIK and O. A. LADYZHENSKAYA, Boundary-value

problems for partial differential equations and certainclasses of operator equations, Uspekhi Matem. Nauk,6 (72) (1956), 41–97; English transl., Amer. Math. Soc.Transl. (2) 10, 1958, 223–281.

[2] O. A. LADYZHENSKAYA, Mixed Boundary-Value Problem forHyperbolic Equation, Gostekhizdat, Moscow, 1953.(Russian)

[3] ——— , The Mathematical Theory of Viscous Incom-pressible Flow, Gostehizdat, Moscow, 1961; Englishtransl., Gordon and Breach, New York-London, 1963.

[4] O. A. LADYZHENSKAYA and N. N. URAL’TSEVA, Linear andQuasilinear Equations of Elliptic Type, Nauka, Moscow,1964 (2nd. ed.), 1973; English transl., Academic Press,New York-London, 1968.

[5] O. A. LADYZHENSKAYA, V. A. SOLONNIKOV, and N. N.URAL’TSEVA, Linear and Quasilinear Equations of Para-bolic Type, Nauka, Moscow, 1967; English transl.,Translations of Mathematical Monographs, 23, Amer.Math. Soc., Providence, RI, 1968.

[6] O. A. LADYZHENSKAYA, Attractors for Semigroups and Evo-lution Equations, Lezioni Lincei, 1988, Cambridge Univ.Press., Cambridge, 1991.

Peter Lax, Cathleen Morawetz, and Louis Nirenberg

Young mathematicians today would have a hardtime imagining how thoroughly the Iron Curtain iso-lated the Soviet Union from the rest of the worldin the 1940s and 1950s. Mathematics was no ex-ception. We were generally aware who the leading

figures were—Gelfand, Kolmogorov, Petrovsky,Pontryagin, Sobolev, Vinogradov—and knew thenames of some of the up-and-coming new postwargeneration—Faddeev, Arnold, Ladyzhenskaya,Oleinik. We were aware of Gelfand’s spectacularwork on normed rings and were influenced byPetrovsky’s survey article on partial differentialequations. But in general we were woefully igno-rant—few of us knew Russian, or even the Cyrillicalphabet; personal encounters were highly re-stricted.

The thaw started onlyafter Stalin died; a keyevent was Khrushchev’s de-nunciation of Stalin’scrimes in 1956, the year theSoviet Government reha-bilitated Ladyzhenskaya’sfather, shot as a traitor in1937. Khrushchev’s speechwas meant only for theparty faithful, but it wassoon disseminated gener-ally thanks to the CIA—credit where credit is due.A few visitors were allowedto enter the Soviet Unionand have relatively free ac-cess to Soviet scientists; theimpressions of these visi-tors were eagerly sought.Leray reported that inLeningrad he saw the Her-mitage, Peterhof, and La-dyzhenskaya.

The Iron Curtain worked in both directions;when Ladyzhenskaya first started to work on theNavier-Stokes equation, she was unaware of thework of Leray and Hopf.

A major sign of a thaw was the large Soviet del-egation, Olga among them, at the Edinburgh In-ternational Congress of Mathematicians in 1958. Formany of us this was the beginning of a mathe-matical and personal friendship with Olga. Afterthe close of the Congress, by chance I (PDL) ran intoa group of Soviet mathematicians at the NationalGallery in London. Olga and I started talking andgot separated from the group. Unfortunately Olgadid not remember the name of the hotel where thedelegation was staying, so we were forced to go tothe Soviet Consulate; with their help the lost sheepwas reunited with her flock.

The trickle of Western visitors turned to a tidein the 1960s. An outstanding event was the Open-ing Conference of the University and Science Cityat Novosibirsk in 1963, under the direction ofLavrentiev. Here friendships started at Edinburghwere renewed under much more relaxed condi-tions. LN recalls that Olga had arranged a sailing

Peter Lax is professor emeritus at the Courant Institute ofthe Mathematical Sciences, New York University. His emailaddress is lax@cims.nyu.edu.

Cathleen Morawetz is professor emeritus at the CourantInstitute of the Mathematical Sciences, New York Univer-sity. Her email address is morawetz@cims.nyu.edu.

Louis Nirenberg is professor emeritus at the Courant In-stitute of the Mathematical Sciences, New York University.His email address is nirenl@cims.nyu.edu.

1330 NOTICES OF THE AMS VOLUME 51, NUMBER 11

party on the Ob Sea on a vessel with a red sail, madeavailable by Sobolev, director of the MathematicsInstitute at Novosibirsk.

The International Congress of Mathematiciansin Moscow in 1966 was another outstanding occa-sion for Western and Soviet mathematicians to gettogether.

Western publishers started putting out Englishtranslations of important books that originally ap-peared in Russian, including Olga’s MathematicalTheory of Viscous Incompressible Flow and her bookwith Ural’tseva, Linear and Quasilinear Elliptic Equa-tions. The AMS performed an immensely usefulservice with its vast translation program of articlesand books, including Olga’s book with Ural’tsevaand Solonnikov on parabolic equations.

Olga’s first work was to use the method of fi-nite differences to solve the Cauchy problem forhyperbolic equations, giving a new proof of Petro-vsky’s result—Petrovsky’s method is incompre-hensible.This work of Olga’s is little known in theWest; Olga herself did not return to it, very likelybecause she decided (as did Friedrichs) that a pri-ori estimates and the methods of functional analy-sis are more effective tools. She did, however, de-vise numerical schemes for solving theNavier-Stokes equation.

Olga consistently used the concept of weak so-lutions, a point of view championed by Friedrichs,whom Olga admired. She was in the forefront ofthe upsurge of interest in elliptic equations. In avery early work she showed that second-order el-liptic boundary-value problems have square inte-grable solutions up to the boundary under very gen-eral boundary conditions. Her books with Ural’tsevaand Solonnikov contain many deep results con-cerning estimates for solutions of elliptic and par-abolic equations. They have greatly extended ideasof Serge Bernstein and techniques of DiGiorgi,Moser, and Nash. These books are basic sources forthese subjects.

The work for which Olga will be rememberedlongest is on the Navier-Stokes equation. This is atechnically very difficult field in which every ad-vance, even modest, requires great effort. One ofthe goals is to decide if the initial value problemin three dimensions has a smooth solution for alltime, and if not, whether a generalized solution isuniquely determined by the initial data. But evenif at some future time these questions are decided,and the Clay Foundation rewards its solver with oneof its million-dollar prizes, the main task of hy-drodynamics remains to deduce the laws govern-ing turbulent flow.

In additional to a large body of technical re-sults, Olga boldly proposed a modification of theNavier-Stokes equations in regions where the ve-locity fluctuates rapidly. She also made importantcontributions to the theory of finite-dimensional

attractors of solutions of the Navier-Stokes equa-tions. Flow-invariant measures on manifolds ofsuch attractors might be a building block of tur-bulence.

On her last visit abroad, at a conference held inMadeira in June 2003, she gave a spirited personalaccount of her work on the Navier-Stokes equation.

For decades, Olga ran a seminar on partial dif-ferential equations that kept up with the develop-ment of the subject worldwide; it had a great in-fluence in the Soviet Union.

Olga’s career shows that even in the darkestdays of Soviet totalitarianism there were coura-geous academics, Petrovsky and Smirnov in Olga’scase, who were willing to defy official proscriptionof the daughter of a man shot as a traitor, perceiveher ability, and ease her path so her talent couldbloom and gain recognition.

It is to the great credit of the international math-ematical community that at the height of the ColdWar, when both sides were piling thousands of nu-clear weapons on top of each other, Soviet andWestern mathematicians formed an intimate fam-ily, where scientific and personal achievementswere admired independently of national origin.Olga was among the most admired, for her courage,for overcoming enormous obstacles, for support-ing those under attack, for her mathematicalachievements, and for her overwhelming beauty.

AcknowledgementPhotographs and the mathematical family tree

are reproduced from the first two volumes, Non-linear Problems in Mathematical Physics and RelatedTopics. In Honor of Professor O. A. LadyzhenskayaI, II, M. Sh. Birman, S. Hildebrandt, V. A. Solon-nikov, N. N. Ural’tseva, eds., of the InternationalMathematical Series published by Kluwer/PlenumPublishers (English) and by Tamara Rozhkovskaya(publisher, Russian). These photos are reproducedhere with permission of the publishers of the In-ternational Mathematical Series.

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