## Ladyzhenskaya, Olga Alexandrovna

## Ladyzhenskaya, Olga Alexandrovna

# LADYZHENSKAYA, OLGA ALEXANDROVNA

(*b*. Kologriv, U.S.S.R., 7 March 1922; *d*. St. Petersburg, Russia, 12 January 2004),

*mathematics, partial differential equations*.

Ladyzhenskaya was one of the very few outstanding female mathematicians of the twentieth century. The general theory of partial differential equations, governing fluids, gases, elasticity, electromagnetism, and quantum physics was developed during the twentieth century. Ladyzhenskaya was a major figure in the treatment of parabolic (typified by u_{t} – u_{xx}= 0) and elliptic (u_{xx} + u_{yy} = 0) equations. She obtained pioneering results in the spectral theory of general elliptic operators and in diffraction. With her student, Nina Ural’tseva, she analyzed in depth the regularity of quasilinear elliptic equations and with Ural’tseva and Vsevolod A. Solonnikov, the regularity of parabolic equations.

**Life and Career** . Ladyzhenskaya grew up in Kologriv in European Russia, where her father was the high school principal and a teacher of mathematics and art. Ladyzhenskaya’s mathematical education began in the Kologriv high school. After school, her father gave her lessons at home. Her father was denounced to the Soviet authorities in 1937, declared an enemy of the people, imprisoned, and then executed. Ladyzhenskaya, unlike her older sisters, was permitted to finish high school, from which she graduated in 1939, but she was not admitted to Leningrad University because of her father. (He was completely exonerated after Khrushchev’s secret speech in 1956, three years after Stalin’s death.) Her mother, Anna Mikhailovna, had to struggle to keep the family housed and fed.

Olga attended a pedagogical institute, taught school in Kologriv after the German invasion of June 1941, and was finally admitted to Moscow University in 1943. She attended one of Israel Gelfand’s first seminars with Mark

Visik and Olga Oleinik, who later became very well known mathematicians and professors at Moscow University. Gelfand’s seminar later became the most renowned of Russian mathematical seminars. All three of them did their student theses with Ivan Petrovsky. Ladyzhenskaya remained close to Visik but had a long rivalry with Oleinik. She graduated in 1947, moved to Leningrad, and was married briefly to her fellow student, Andrei Kiselev. She completed her PhD with Sergei Sobolev and Vladimir Smirnov in 1951, and defended her DSc at Moscow University in 1953. She began teaching in the Physics Department of the University of Leningrad in 1950; she became a member of the Mathematical Physics Laboratory of the Leningrad branch of the Steklov Institute of Mathematics starting in 1954 and was made head of the laboratory in 1961.

**Mathematical Work** . Ladyzhenskaya’s university thesis, under the supervision of Ivan Petrovsky, was a result on the approximation of hyperbolic (u _{tt}- u _{xx}= 0) equations, but her earliest important success was her elegant estimates for elliptic operators involving spaces of generalized functions. Her best-known “sharp” estimate is
. Here *u* is a function defined in a domain on whose boundary *u*= 0 and Λ is an elliptic operator acting on *u*. For example, *u* is an electric potential and Λ *u* is the charge density, the operator Λ is the Laplacian,
measures the function by the mean square of its value on the domain *W*, and measures
the mean square of all the derivatives up to order 2. From such estimates one can prove uniqueness and existence for solutions of equations involving Λ. “Sharp” means that the constant *C* can be chosen so that the inequality is an equality for some function *u* for a given domain *W*.

For Navier-Stokes equations (incompressible viscous flow), Ladyzhenskaya proved unique global solvability for the two-dimensional case, refining the earlier work of Jean Leray in “Sur le movement d’un liquide visqueux emplissant l’espace” (1934; On the motion of a viscous fluid filling space) and Eberhard Hopf in “Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen” (1951; On the initial value problem for the fundamental equations of fluid dynamics). For the three-dimensional case, Ladyzhenskaya introduced innovative modifications to Navier-Stokes equations to treat large velocity fluctuations. She also established the first “attractor” results. This means that a solution of an initial value problem, under certain restrictions, “eventually” is very close to a particular solution, “the attractor,” where it stays. Here she was one of the important innovators.

**Legacy and Honors** . Ladyzhenskaya authored or coauthored seven books and 250 papers. Her most influential works were her books, which contain many original theorems.

Her honors include a corresponding membership in the Academy of Sciences U.S.S.R. in 1981; and full membership in the Russian Academy of Sciences in 1991. She was made a foreign member of the Lincei National Academy in Rome in 1989 and a member of the American Academy of Arts of Sciences in 2001. She received an honorary doctorate from the University of Bonn in 2002. In the latter year Ladyzhenskaya also received the Lomonosov Medal of the Russian Academy of Sciences. She gave the Noether lecture at the 1994 congress of the International Mathematical Union.

Noted everywhere for her great charm, beauty, culture, and depth of feeling, Olga Ladyzhenskaya several times risked herself and her career to help individuals oppressed by the Soviet regime.

A number of her students, in particular Ludwig Faddeev, Nina Ural’tseva, and Vsevolod A. Solonnikov have made major contributions to physics, to partial differential equations, and to the Navier-Stokes equations.

## BIBLIOGRAPHY

*For a complete bibliography of Ladyzhenskaya’s writings, see G. A. Seregin and N. N. Ural’tseva, “Ol’ga Aleksandrovna Ladyzhenskaya (on her 80th birthday).”* Russian Mathematical Surveys *58 (2003): 395–425*.

### WORKS BY LADYZENSKAYA

With N. N. Ural’tseva. *Lineinye i kvazilineinye uravneniia ellipticheskogo tipa*. Moscow, USSR: Nauka, 1964. Translated by Scripta Technica as *Linear and Quasilinear Equations of Elliptic Type* (New York: Academic Press, 1968).

With V. A. Solonnikov and N. N. Ural’tseva. *Lineinye i kvazilineinye uravneniia parabolicheskogo tipa*. Moscow, USSR: Nauka, 1967. Translated by S. Smith as *Linear and Quasilinear Equations of Parabolic Type* (Providence, RI: American Mathematical Society, 1968).

### OTHER WORKS

Friedlander, Susan, Peter Lax, Cathleen Morawetz, et al. “Olga Alexandrovna Ladyzhenskaya.” *Notices of the American Mathematical Society* 51, no. 11 (December 2004): 1320–1331. Hopf, Eberhard. “Uber die Anfangswertaufgabe fur die hydrodynamischen Grundgleichungen.” *Mathematische Nachrichten* 4 (1951): 213–231.

Leray, Jean. “Sur le movement d’un liquide visqueux emplissant l’espace.” *Acta Mathematica* 63 (1934): 193–248.

Struwe, Michael. “Olga Ladyzhenskaya—a Lifelong Devotion to Mathematics.” In *Geometric Analysis and Nonlinear Partial Differential Equations*, edited by Stefan Hildebrandt and Hermann Karcher, pp. 1–10. Berlin: Springer, 2003.

*Cathleen Synge Morawetz*