# St. Petersburg: Olga Alexandrovna Ladyzhenskaya

Hi, this is Sergey Lupuleac.

Two weeks ago one of my ECMI colleagues asked me about famous mathematicians from St. Petersburg. I emphasize: not from all Russia, but from a Saint Petersburg town. I listed several well-known names from Leonard Euler to Grigori Perelman. But then I thought that it is a very good idea for a series of theme-based posts inside this blog. These posts will be dedicated to our great fellow townsmen: Petersburgers.

The topic of the first post was unexpectedly suggested by Google, that at March 7th published the following Doodle :

It is about Olga Alexandrovna Ladyzhenskaya. In Russia, we always use patronymic names, when addressing someone with respect, and Olga Alexandrovna is one of the most respectful persons I’ve ever known. Yes, I and some of my close colleagues had the honour to know Olga Alexandrovna Ladyzhenskaya personally. While being PhD students, we attended seminars headed by Professor Ladyzhenskaya. Now I give a course of PDE to our students based on her book. I also named my daughter Olga after her.

I asked my colleagues Julia Shinder (who also knew Olga Alexandrovna personally) and Timofey Shilkin (who worked with Olga Alexandrovna in Steklov Institute of Mathematics) to prepare the following material. It is fully based on [1].

Olga Aleksandrovna Ladyzhenskaya was born in the town of Kologriv in the Kostroma region in 1922. Her father, who was the first teacher of Olga Aleksandrovna and who cultivated her interest in exact sciences, was subjected to repression and died in 1937. In 1939 she completed her studies in school with a gold medal but she was not allowed to enter Leningrad State University because she was a daughter of an “enemy of the Russian people.” The only educational institution where she could enter was the Pokrovskii Pedagogical Institute in Leningrad at which she was studying until 1941. In the first years of World War II, she was teaching in her native town, but the desire to learn and a happy chance had brought her to Moscow, where she became a second-year student of Moscow State University at the department of mechanics and mathematics, from which she graduated in 1947 with recommendations to continue the studies as a post-graduate student.

In the same year, she married and went to Leningrad, where she became a post-graduate student of Leningrad State University, at the department of mathematics and mechanics. S. L. Sobolev (yes, Sobolev spaces are named after him) was the scientific adviser of Olga Aleksandrovna.

Academician V. I. Smirnov exerted great influence on the scientific and personal destiny of Olga Aleksandrovna. Together with him, she organized a town seminar, which was oriented to the latest directions in mathematical physics and in the theory of multidimensional boundary-value problems. Vladimir Ivanovich and Olga Aleksandrovna headed this seminar together for a long time. Later on, it was headed by Olga Aleksandrovna and the seminar was named in honour of V. I. Smirnov.

One can recognize several periods in the creative work of Olga Aleksandrovna. The first period includes the papers written until 1953, when she defended the Doctor’s Thesis. In the same year, the first monograph of Olga Aleksandrovna has appeared. The problems that engaged Olga Aleksandrovna at this period came from two Moscow seminars headed by G. I. Petrovskii and I. M. Gelfand, respectively, which she attended when she studied at Moscow State University. At that time, at the seminar headed by Gelfand, the problem concerning efficient description of the domains of definition of the closures in L2(Ω) of operators of elliptic type was discussed actively as a problem number one for partial differential equations. In 1951, in [2] Olga Aleksandrovna proved her famous “second basic inequality” for second-order elliptic operators with smooth coefficients, i.e., the estimate which satisfies any of the classical homogeneous conditions on the boundary of the domain. This inequality furnished a complete and maximally general solution of the above-mentioned problem.

Another achievement of Olga Aleksandrovna at that period was the justification of the Fourier method for hyperbolic equations. This series of papers served as a basis for the first monograph of Olga Aleksandrovna titled “The mixed boundary-value problem for the hyperbolic equation” was published by “Gostekhizdat” in 1953 (see [3]).

Over the period from the late 1950s to the early 1960s, she was invited three times to work at the famous Institute of Advanced Studies in Princeton (USA), and every time the authorities refused the visit. But this did not influence the scientific activity of Olga Aleksandrovna, and she went on working with previous intensity.

The new stage in the scientific biography of Olga Aleksandrovna began in 1954 when she became a scientific researcher and from 1961 the head of the Laboratory of Mathematical Physics of the Leningrad Department of the Steklov Mathematical Institute of the Academy of Sciences of the USSR. For many years, Olga Aleksandrovna combined successful scientific work at LOMI with teaching at the department of physics. At that time, she had the first pupils: O. A. Guseva, V. A. Solonnikiv, N. N. Uraltseva, L. D. Faddeev, K. K. Golovkin,. A. P. Oskolkov, A. V. Ivanov, V. Ya. Rivkind; later L. V. Kapitanskii, V. Shubov, and others.

The list of her papers strikes us by the breadth of her interests at that time. Here there are problems in the abstract theory of operators [4], the general theory of boundary-value problems [5, problems in diffraction theory [6], the convergence of finite difference methods [7], problems of the spectral theory of differential operators [8], the theory of quasilinear equations [9], problems in the calculus of variations [10], and many others. Throughout that period, Olga Aleksandrovna published the first papers on hydrodynamics (see [11]) – the subject matter, which always occupied a central place in her studies over the last fifty years. In 1958, in [12] Professor Olga Ladyzhenskaya established a version of the multiplicative inequalities known at present as Ladyzhenskaya inequalities. These inequalities enable her to prove the global unique solvability of the Navier–Stokes system in the two-dimensional case, see [12] and [13]. In the three-dimensional case, for a finite interval of time the length of which depends on some norms of the data of the problem and for an infinite interval of time if the norms of data are sufficiently small, a similar result was established by Olga Aleksandrovna in collaboration with A. A. Kiselev [14] in 1957.

In [15], she proved the global solvability of the basic boundary-value problem for the stationary Navier–Stokes system in arbitrary bounded domains, as well as the problem on a flow in external domains. In 1961, all these investigations were included in the famous monograph of Olga Aleksandrovna entitled “Mathematical problems in the dynamics of viscous incompressible liquids” [16] translated into many languages. In the history of hydrodynamics, it was the first book that presented the subject consistently and rigorously. Up to the present, it is a consummate introduction to this area.

In [14], it was suggested in the firm belief that the class of solutions found by Hopf [17] in 1951 is impermissibly wide in the sense that, in the three-dimensional case, the uniqueness theorem is not valid in it (for this reason, Olga Aleksandrovna called these solutions weak). Later on, she confirmed this point of view. These results, as well as the results of Ladyzhenskaya on finite difference schemes and some information on the modifications of Navier–Stokes equations proposed by her, were included in the second Russian edition of monograph [16] that has been published in 1970 (see [18]).

Another problem, which was a central subject of study by Olga Aleksandrovna beginning from the middle 50s, is the regularity theory for solutions of quasilinear equations of elliptic and parabolic types. Most of the results in this direction were obtained by Olga Aleksandrovna in collaboration with her pupil N. N. Uraltseva.

The starting points of this research were paper [9] written by Olga Aleksandrovna in 1956 on the estimation of the gradients of the solutions of elliptic and parabolic quasilinear equations and famous paper [19] by E. De Giorgi who established in 1957 that the solutions of the linear uniformly elliptic equation with measurable coefficients are H¨older continuous. (A similar result was obtained independently by J. Nash [20] in 1958.) Olga Aleksandrovna Ladyzhenskaya and Nina Nikolaevna Uraltseva developed considerably the technique of De Giorgi, extending it to nonhomogeneous linear and quasilinear equations of elliptic and parabolic types. In addition, she devised a technique for proving a priori estimates of solutions of elliptic equations with strong nonlinearities. These investigations enable her to obtain the best possible results concerning the regularity of solutions of quasilinear equations that satisfy the so-called natural conditions for growth.

Thus, a complete and ultimate solution was found for the 19th and 20th Hilbert problems in the scalar case. In 1964, these investigations were presented in monograph [21], the second edition of which has appeared in 1973. In 1967, similar results for quasilinear parabolic equations were included in monograph [22] written by Professor Ladyzhenskaya in collaboration with V. A. Solonnikov and N. N. Uraltseva. A series of joint papers of Olga Aleksandrovna and Nina Nikolaevna devoted to nonlinear parabolic equations were awarded by the USSR State Premium in 1969.

Later on, Olga Aleksandrovna turns to the study of nonuniformly elliptic and parabolic quasilinear equations. In [23] (joint with N. N. Uraltseva), local estimates of the maximum of the modulus of the gradient of the solutions of equations of the mean curvature type were established.

One of the greatest achievements of Olga Aleksandrovna is her contribution to the development of the theory of attractors of infinite-dimensional dynamical systems. In the pioneering paper [15] written in 1972, she proved the existence of a global B-attractor for the Navier–Stokes system in the two-dimensional case. This result came to the attention not only of mathematicians but of physicist-theorists as well. We also note paper [22] in which there is a simple and very elegant way for estimating the Hausdorff dimension of the attractor for dissipative systems generated by partial differential equations of various types. Many of the investigations carried out by Olga Aleksandrovna in this direction were included in her monograph [26] published in Cambridge in 1991. This book was awarded by the Kovalevskaya Premium of the Russian Academy of Sciences in 1992.

In the 90s years, Olga Aleksandrovna continues working successfully in various areas of mathematical physics. At that time, she resumed her research in the modified Navier–Stokes systems (see [27, 28]) – models of hydrodynamic type, which she proposed at the World Congress of Mathematics in 1965 for the description of motion of a viscous liquid for large gradients of velocities. At that time, Olga Aleksandrovna actively studied the theory of fully nonlinear equations [29–31] and many other problems.

The scientific achievements of Olga Aleksandrovna are generally recognized. Professor Olga Ladyzhenskaya was a Full Member of the Russian Academy of Sciences since 1990 (a Corresponding Member since 1981). It is interesting to note that she won the “official” recognition in the West earlier than in Russia, although in fact, she did not go abroad until 1988. Moreover, she is a Foreign Member of Deutsche Academia der Naturforschung Leopoldine since 1985 and of Academia Nazionale dei Lincei since 1989. In 2002, she became a Foreign Member of the American Academy of Sciences and Arts in Berkeley and an Honorary Doctor of Bonn University.

REFERENCES

- A. A. Archipova, M. S. Birman, V. S. Buslaev, V. G. Osmolovskii, S. I. Repin, G. A. Seregin, N. N. Uraltseva, T. N. Shilkin, “To Olga Aleksandrovna Ladyzhenskaya on the occasion of her jubilee”, Journal of Mathematical Sciences, Vol. 123, No. 6, 2004
- O. A. Ladyzhenskaya, “On the closure of the elliptic operator,” Dokl. Akad. Nauk SSSR, 79, No. 5, 723–725 (1951).
- O. A. Ladyzhenskaya, The Mixed Boundary-Value Problem for the Hyperbolic Equation [in Russian], Moscow (1953).
- O. A. Ladyzhenskaya, “On the solution of nonstationary operator equations of various types,” Dokl. Akad. Nauk SSSR, 102, No. 2, 207–210 (1955).
- O. A. Ladyzhenskaya, “On the solvability of the basic boundary-value problems for equations of parabolic and hyperbolic types,” Dokl. Akad. Nauk SSSR, 96, No. 3, 395–397 (1954).
- O. A. Ladyzhenskaya, “On the solution of the general problem in diffraction,” Dokl. Akad. Nauk SSSR, 96, No. 3, 433–436 (1954).
- O. A. Ladyzhenskaya, “The finite difference method in the theory of partial differential equations,” Usp. Mat. Nauk, 12(5), 123–148 (1957).
- O. A. Ladyzhenskaya and L. D. Faddeev, “To the theory of perturbations of the continuous spectrum,” Dokl. Akad. Nauk SSSR, 120, No. 6, 1187–1190 (1958).
- O. A. Ladyzhenskaya, “The first boundary-value problem for quasilinear parabolic equations,” Dokl. Akad. Nauk SSSR, 107, No. 5, 636–639 (1956).
- O. A. Ladyzhenskaya, “On differential properties of the generalized solutions of some multidimensional variational problems,” Dokl. Akad. Nauk SSSR, 120, No. 5, 956–959 (1958).
- A. A. Kiselev and O. A. Ladyzhenskaya, “On the solution of linearized equations of plane nonstationary flow of a viscous incompressible liquid,” Dokl. Akad. Nauk SSSR, 95, No. 6, 1161–1164 (1954).
- O. A. Ladyzhenskaya, “Solution “in the large” of the boundary-value problem for Navier–Stokes equations in the case of two space variables,” Dokl. Akad. Nauk SSSR, 123, No. 3, 427–429 (1958).
- O. A. Ladyzhenskaya, “Solution “in the large” of the nonstationary boundary-value problem for Navier–Stokes system with two space variables,” Comm. Pure Appl. Math., 12(3), 427–433 (1959).
- A. A. Kiselev and O. A. Ladyzhenskaya, “The existence and uniqueness of a solution of the nonstationary problem for a viscous incompressible liquid,” Izv. Akad. Nauk SSSR, Ser. Mat., 21(5), 665–680 (1957).
- O. A. Ladyzhenskaya, “The study of Navier–Stokes equations for stationary motion of an incompressible liquid,” Usp. Mat. Nauk, 15, 75–97 (1959).
- O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of Viscous Incompressible Liquids [in Russian], Moscow (1961).
- E. Hopf, “¨Uber die Anfangswertaufgabe f¨ur die hydrodynamischen Grundgleichungen,” Math. Nachrichten, 4, 213–231 (1950–1951).
- O. A. Ladyzhenskaya, Mathematical Problems in the Dynamics of Viscous Incompressible Liquids [in Russian], 2nd ed., Moscow (1970).
- E. De Giorgi, “Sulla differenziabilita e l’analitica delle estremali degli integrali multipli regolari,” Memorie delle Acc. Sci. Torino, Ser. 3, 3(1), 25–43 (1957).
- J. Nash, “Continuity of solutions of parabolic and elliptic equations,” Amer. J. Math., 80 (4), 931–954 (1958).
- O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Equations of Elliptic Type [in Russian], Nauka, Moscow (1964).
- O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).
- O. A. Ladyzhenskaya and N. N. Uraltseva, “Local estimates for gradients of solutions of nonuniformly elliptic and parabolic equations,” Comm. Pure Appl. Math., 23, 677–703 (1970).
- O. A. Ladyzhenskaya, “On the dynamical system generated by Navier–Stokes equations,” Zap. Nauchn. Semin. LOMI, 27, 91–115 (1972).
- O. A. Ladyzhenskaya, “On the finite dimensionality of bounded invariant sets for Navier–Stokes systems and other dissipative systems,” Zap. Nauchn. Semin. LOMI, 115, 137–155 (1982).
- O. A. Ladyzhenskaya, Attractors for Semi-Groups and Evolution Equations, Cambridge Univ. Press, Cambridge (1991).
- O. A. Ladyzhenskaya and G. A. Seregin, “Smoothness of solutions of systems describing the flow of generalized Newtonian liquids and estimation of dimensions of their attractors,” Izv. Ross. Akad. Nauk, Ser. Mat., 62, No. 1, 59–122 (1998).

- O. A. Ladyzhenskaya and G. A. Seregin, “On disjointness of solutions to the MNS equations,” Trans. Amer. Math. Soc., 189, 159–179 (1999).
- N. M. Ivochkina and O. A. Ladyzhenskaya, “Estimation of the second-order derivatives on the boundary for hypersurfaces varying under the action of their principal curvatures,” Algebra Analiz, 9, 30–50 (1997).
- N. M. Ivochkina and O. A. Ladyzhenskaya, “Estimation of the first-order derivatives for the solutions of some classes of nonlinear parabolic equations,” Algebra Analiz, 9, 109–131 (1997).
- N. M. Ivochkina and O. A. Ladyzhenskaya, “On classical solvability of the first initial boundary-value problem for equations generated by curvatures,” TMNA, J. Juliusz Schauder Center, 11, 375–395 (1998).