Eberhard Hopf

Eberhard Hopf was an Austrian-American mathematician who, in 1951, independently of Leray and more rigorously, proved the existence of weak solutions to the Navier–Stokes equations in bounded domains with no-slip boundary conditions. His work, published in 1951, completed the formation of the theory of weak solutions, now known as Leray–Hopf solutions. He used the Galerkin method and proved that for any initial velocity field with finite kinetic energy there exists a global-in-time weak solution satisfying the energy inequality. This inequality plays a key role in the analysis of possible singularity: if the solution loses smoothness, the energy must behave in a certain way, which is used in modern research on finite-time blow-up. Hopf is also known for his work in ergodic theory, topological dynamics, and differential geometry, but for the Navier–Stokes problem his contribution consists in the rigorous justification of weak solutions and energy estimates, which are the foundation for all subsequent attempts to prove or disprove regularity.