estimates for the Boltzmann collision operator, thereby finding compactness results for solutions of the Boltzmann equation. As a particular application of his compactness theory, he was able to show that solutions subsequentially converge at infinite time to Maxwell distributions. DiPerna and Lions also established a similar result for the Maxwell−Vlasov equations. Viscosity solutions Michael Crandall and Lions introduced the notion of viscosity solution, which is a kind of generalized solution of Hamilton–Jacobi equations. Their definition is significant since they were able to establish a well-posedness theory in such a generalized context. The basic theory of viscosity solutions was further worked out in collaboration with Lawrence Evans. Using a min-max quantity, Lions and Jean-Michel Lasry considered mollification of functions on Hilbert space which preserve analytic phenomena. Their approximations are naturally applicable to Hamilton-Jacobi equations, by regularizing sub- or super-solutions. Using such techniques, Crandall and Lions extended their analysis of Hamilton-Jacobi equations to the infinite-dimensional case, proving a comparison principle and a corresponding uniqueness theorem. Crandall and Lions investigated the numerical analysis of their viscosity solutions, proving convergence results both for a finite difference scheme and artificial viscosity. The comparison principle underlying Crandall and Lions' notion of viscosity solution makes their definition naturally applicable to second-order elliptic partial differential equations, given the maximum principle. Crandall, Ishii, and Lions' survey article on viscosity solutions for such equations has become a standard reference work. Mean field games With Jean-Michel Lasry, Lions has contributed to the development of mean-field game theory. Major publications Articles. Textbooks. References External links College de France his resume at the Collège de France website (in French) 1956 births Living people People from Grasse Academic staff of the Collège de France 20th-century French mathematicians 21st-century French mathematicians Fields Medalists Mathematical analysts École Normale Supérieure alumni Lycée Louis-le-Grand alumni Members of the French Academy of Sciences PDE theorists International Mathematical Olympiad participants Academic staff of Nancy-Université Prix Paul Doistau–Émile Blutet laureates University of Chicago staff