Jean Leray

Jean Leray was a French mathematician who in 1934 constructed global-in-time weak solutions of the Navier–Stokes equations for incompressible fluids in the whole space and in bounded domains. He introduced the concept of a "turbulent solution", now called a Leray–Hopf weak solution. His approach is based on approximating the equations, obtaining a priori energy estimates, and passing to the limit, which allows proving the existence of a solution for any finite time interval. However, the question of uniqueness and regularity of such solutions in the three-dimensional case remains open: it is unknown whether a solution can lose smoothness or even blow up in finite time. This very question is one of the Millennium Problems. Leray also made fundamental contributions to algebraic topology, creating sheaf theory and spectral sequences, but in the context of the Navier–Stokes equations his main achievement is the formulation of the global regularity problem and the construction of weak solutions, which serve as the foundation for modern research, including the analysis of possible finite-time blow-up.