Pierre-Louis Lions

fl. 1956, Grasse, France

Differential equations and dynamical systemsAnalysisApplied and computational mathematicsFields Medal 1994

French analyst who created viscosity-solution theory and the concentration-compactness method, and proved existence for the Boltzmann equation; Fields Medal 1994.

Pierre-Louis Lions was born in Grasse in 1956, the son of the mathematician Jacques-Louis Lions. After studying at the École Normale Supérieure he took his doctorate under Haïm Brézis in 1980 and rose quickly to a professorship at the Université Paris-Dauphine.

His Fields Medal (Zürich, 1994) crowned a decade that reshaped nonlinear PDE: the viscosity-solution theory for Hamilton–Jacobi equations developed with Michael Crandall; the concentration-compactness principle for variational problems; and, with Ronald DiPerna, the first existence theorem for renormalized solutions of the Boltzmann equation. His mathematical range runs from the mean curvature flow and image processing to stochastic control and mathematical finance.

In the 2000s, with Jean-Michel Lasry, Lions founded mean field game theory — a model of strategic interaction among a continuum of agents coupling a Hamilton–Jacobi equation with a Fokker–Planck equation — which has become a standard tool in economics and engineering. He holds a chair at the Collège de France, has been awarded the Grand Prix Ampère and the Prize of the Japan Academy, and is known for applying PDE methods to industrial problems.

Workplaces: Collège de France, École Polytechnique

France

Award-winning work

Fields Medal 1994Lions was honored for his analysis of nonlinear partial differential equations. With Michael Crandall he created the theory of viscosity solutions — a notion of weak solution for Hamilton–Jacobi equations and fully nonlinear second-order equations that is stable under limits and admits comparison principles, becoming the standard framework for optimal control, geometric flows and finance. His concentration-compactness method solved variational problems that lose compactness, by classifying all ways a minimizing sequence can fail to converge. With Ronald DiPerna he proved the global existence of renormalized solutions of the Boltzmann equation, the first rigorous existence result of any kind for that fundamental kinetic equation.

Contributions, linked to the library

Great problemNavier–Stokes existence and smoothnessHis work on weak solutions of fluid equations and kinetic theory bears directly on the Navier–Stokes regularity problem, one of the central open questions of nonlinear PDE.Library topicSystems of differential equationsCreator of viscosity solutions and concentration-compactness, two of the main tools of modern PDE theory.