Louis Nirenberg

The Navier-Stokes equations were developed in the early 1800s to model the physics of fluid mechanics.

In his doctoral work, he solved the "Weyl problem" in differential geometry, which had been a well-known open problem since 1916.

Following his doctorate, he became a professor at the Courant Institute, where he remained for the rest of his career.

He was the advisor of 45 PhD students, and published over 150 papers with a number of coauthors, including notable collaborations with Henri Berestycki, Haïm Brezis, Luis Caffarelli, and Yanyan Li, among many others.

He continued to carry out mathematical research until the age of 87.

Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.

Nearly all of his work was in the field of partial differential equations.

Many of his contributions are now regarded as fundamental to the field, such as his strong maximum principle for second-order parabolic partial differential equations and the Newlander-Nirenberg theorem in complex geometry.

He is regarded as a foundational figure in the field of geometric analysis, with many of his works being closely related to the study of complex analysis and differential geometry.

Nirenberg was born in Hamilton, Ontario to Ukrainian Jewish immigrants.

Jean Leray, in a seminal achievement in the 1930s, formulated an influential notion of weak solution for the equations and proved their existence.

His work was later put into the setting of a boundary value problem by Eberhard Hopf.

In the 1930s, Charles Morrey found the basic regularity theory of quasilinear elliptic partial differential equations for functions on two-dimensional domains.

Nirenberg, as part of his Ph.D. thesis, extended Morrey's results to the setting of fully nonlinear elliptic equations.

The works of Morrey and Nirenberg made extensive use of two-dimensionality, and the understanding of elliptic equations with higher-dimensional domains was an outstanding open problem.

The Monge-Ampère equation, in the form of prescribing the determinant of the hessian of a function, is one of the standard examples of a fully nonlinear elliptic equation.

He attended Baron Byng High School and McGill University, completing his BS in both mathematics and physics in 1945.

Through a summer job at the National Research Council of Canada, he came to know Ernest Courant's wife Sara Paul.

She spoke to Courant's father, the eminent mathematician Richard Courant, for advice on where Nirenberg should apply to study theoretical physics.

Following their discussion, Nirenberg was invited to enter graduate school at the Courant Institute of Mathematical Sciences at New York University.

In 1949, he obtained his doctorate in mathematics, under the direction of James Stoker.

The study of the BMO function space was initiated by Nirenberg and Fritz John in 1961; while it was originally introduced by John in the study of elastic materials, it has also been applied to games of chance known as martingales.

A breakthrough came with work of Vladimir Scheffer in the 1970s.

He showed that if a smooth solution of the Navier−Stokes equations approaches a singular time, then the solution can be extended continuously to the singular time away from, roughly speaking, a curve in space.

Without making such a conditional assumption on smoothness, he established the existence of Leray−Hopf solutions which are smooth away from a two-dimensional surface in spacetime.

Such results are referred to as "partial regularity."

Soon afterwards, Luis Caffarelli, Robert Kohn, and Nirenberg localized and sharpened Scheffer's analysis.

The key tool of Scheffer's analysis was an energy inequality providing localized integral control of solutions.

It is not automatically satisfied by Leray−Hopf solutions, but Scheffer and Caffarelli−Kohn−Nirenberg established existence theorems for solutions satisfying such inequalities.

With such "a priori" control as a starting point, Caffarelli−Kohn−Nirenberg were able to prove a purely local result on smoothness away from a curve in spacetime, improving Scheffer's partial regularity.

Similar results were later found by Michael Struwe, and a simplified version of Caffarelli−Kohn−Nirenberg's analysis was later found by Fang-Hua Lin.

In an invited lecture at the 1974 International Congress of Mathematicians, Nirenberg announced results obtained with Eugenio Calabi on the boundary-value problem for the Monge−Ampère equation, based upon boundary regularity estimates and a method of continuity.

His 1982 work with Luis Caffarelli and Robert Kohn made a seminal contribution to the Navier–Stokes existence and smoothness, in the field of mathematical fluid mechanics.

Other achievements include the resolution of the Minkowski problem in two-dimensions, the Gagliardo–Nirenberg interpolation inequality, the Newlander-Nirenberg theorem in complex geometry, and the development of pseudo-differential operators with Joseph Kohn.

In 2014, the American Mathematical Society recognized Caffarelli−Kohn−Nirenberg's paper with the Steele Prize for Seminal Contribution to Research, saying that their work was a "landmark" providing a "source of inspiration for a generation of mathematicians."

The further analysis of the regularity theory of the Navier−Stokes equations is, as of 2021, a well-known open problem.

On January 26, 2020, Nirenberg died at the age of 94.

Nirenberg's work was widely recognized, including the following awards and honors:

Nirenberg is especially known for his collaboration with Shmuel Agmon and Avron Douglis in which they extended the Schauder theory, as previously understood for second-order elliptic partial differential equations, to the general setting of elliptic systems.

With Basilis Gidas and Wei-Ming Ni he made innovative uses of the maximum principle to prove symmetry of many solutions of differential equations.