Olga Ladyzhenskaya He was one of the most influential and brilliant people of the 20th century in the field of partial differential equations (PDEs). She was born on March 7, 1922, just 100 years ago now, in Kologriv, Russia, into a family with noble origins. Her background caused her great difficulties, both personally and professionally, in a Soviet Union led by Stalin. Her father was considered an enemy of the nation and executed when Olga was 15 years old.
Despite the difficulties, Ladyzhenskaya’s talent for mathematics and teaching allowed her to reach Moscow State University in 1943, after being denied access to Leningrad University in 1939. In Moscow, she came into contact with prestigious mathematicians such as Ivan Petrovsky, Vyacheslaw Stepanov, Andrei Tikhonov or Ilia Vekua, who sparked her interest in PDEs and mathematical physics, especially Nikolai Smirnov.
Ladyzhenskaya completed her doctoral thesis in 1951, but was unable to publish it until 1953, after Stalin’s death. In it she studied the existence of solutions of certain types of EDP problems called hyperbolic. To do this, she adapted the so-called finite difference method, which today is an essential tool for theoretical and applied mathematics. Indeed, many of the methods used to find approximate solutions of equations using a computer use it.
Along the same lines, he introduced fundamental methods to understand equations in which second-order elliptic operators appear, very frequent in physical systems of a certain complexity, such as a material subjected to forces that attempt to deform it. The ideas he developed are now part of theories such as operator spectral analysis and have contributed to the establishment of notions such as the “weak solution”, which is an essential tool of modern EDP theory. These works contributed to the resolution of problem 20 of the list of 23 that in 1900 the German mathematician David Hilbert proposed as the most important challenges for mathematics in the 20th century.
The rest of his career was devoted to the study of the existence and uniqueness of solutions of some of the most important PDEs in physics, such as the elasticity equations, which model the behavior of materials, the Schrödinger equation, that governs the quantum world or Maxwell’s, which describes electromagnetism. But, without a doubt, the equations that most interested Olga were those of Navier-Stokes. These equations, proposed independently by Claude-Louis Henri Navier in 1822 and by George Gabriel Stokes in 1845, describe the motion of an incompressible fluid, building on the earlier work of Leonhard Euler.
However, it was not until the middle of the 20th century that Jean Leray and Eberhard Hopf built a theory of the existence of solutions for them, giving rise to the so-called Leray-Hopf solutions. The problem of the regularity of these solutions and their uniqueness are still open today and are one of the central issues in contemporary mathematical research. The Clay Institute of Mathematics offers a million dollars for the solution of the first. Some of the most profound results on these problems are due to Ladyzhenskaya and one of the classic books on this discipline, The Mathematical Theory of Viscous Incompressible Flowit is your work.
The first results he obtained in fluid mechanics were for the Stokes system, which considers the Navier-Stokes equations independent of time. His main achievement was to demonstrate the existence of solutions in bounded domains in the fifties of the last century. Along with Vsevolod Solonnikov, he would later prove his uniqueness as well. Later, Ladyzhenskaya proved the local existence of solutions for the Navier-Stokes equations and their uniqueness, under more restrictive conditions than those that the Leray-Hopf equations satisfy, in general.
In 1967 he proved one of his most powerful results. In it, he showed that Leray-Hopf solutions are regular (smooth) if some quantity, which measures the size of the velocity, remains finite. One way to understand this theorem, known as the Ladyzhenskaya-Prodi-Serrin criterion –since Giovanni Prodi and James Serrin independently proved the same result, ignoring certain technical issues, on the same dates–, is the following: if the The movement of a fluid ceases to be regular, that is, if it undergoes sudden changes, it is because the speed has become enormous. There are no intermediate cases, we cannot see a sharp change in the direction of its motion if its speed is finite.
This theorem, and the arguments used to prove it, are still being explored today to understand the behavior of the solutions of the Navier-Stokes equations.
Olga Ladyzehnskaya died on January 12, 2004 at the age of 81. Until her death, she remained active investigating the solutions of the PDEs.
Angel Castro is a researcher at ICMAT.
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems”.
Edition and coordination: Agate A. Timón G Longoria (ICMAT).
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