In 1934 Leray published an important paper that founded the study of weak solutions of the Navier–Stokes equations.

[2] In the same year, he and Juliusz Schauder discovered[3] a topological invariant, now called the Leray–Schauder degree, which they applied to prove the existence of solutions for partial differential equations lacking uniqueness.

His main work in topology was carried out while he was in a prisoner of war camp in Edelbach, Austria from 1940 to 1945.

He concealed his expertise on differential equations, fearing that its connections with applied mathematics could lead him to be asked to do war work.

Leray's work of this period proved seminal to the development of spectral sequences and sheaves.