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. 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. 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. Navier-Stokes equations The Navier-Stokes equations were developed in the early 1800s to model the physics of fluid mechanics. 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. 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 2014, the American Mathematical Society recognized Caffarelli−Kohn−Nirenberg's paper with the Steele Prize for Seminal Contribution