general conditions. Significant "a priori" estimates for solutions were found by Lions in collaboration with Djairo Guedes de Figueiredo and Roger Nussbaum. In more general settings, Lions introduced the "concentration-compactness principle," which characterizes when minimizing sequences of functionals may fail to subsequentially converge. His first work dealt with the case of translation-invariance, with applications to several problems of applied mathematics, including the Choquard equation. He was also able to extend parts of his work with Berestycki to settings without any rotational symmetry. By making use of Abbas Bahri's topological methods and min-max theory, Bahri and Lions were able to establish multiplicity results for these problems. Lions also considered the problem of dilation invariance, with natural applications to optimizing functions for dilation-invariant functional inequalities such as the Sobolev inequality. He was able to apply his methods to give a new perspective on previous works on geometric problems such as the Yamabe problem and harmonic maps. With Thierry Cazenave, Lions applied his concentration-compactness results to establish orbital stability of certain symmetric solutions of nonlinear Schrödinger equations which admit variational interpretations and energy-conserving solutions. Transport and Boltzmann equations In 1988, François Golse, Lions, Benoît Perthame, and Rémi Sentis studied the transport equation, which is a first-order linear partial differential equation. They showed that if the first-order coefficients are randomly chosen according to some probability distribution, then the corresponding function values are distributed with regularity which is enhanced from the original probability distribution. These results were later extended by DiPerna, Lions, and Meyer. In the physical sense, such results, known as velocity-averaging lemmas, correspond to the fact that macroscopic observables have greater smoothness than their microscopic rules directly indicate. According to Cédric Villani, it is unknown if it is possible to instead use the explicit representation of solutions of the transport equation to derive these properties. The classical Picard–Lindelöf theorem deals with integral curves of Lipschitz-continuous vector fields. By viewing integral curves as characteristic curves for a transport equation in multiple dimensions, Lions and Ronald DiPerna initiated the broader study of integral curves of Sobolev vector fields. DiPerna and Lions' results on the transport equation were later extended by Luigi Ambrosio to the setting of bounded variation, and by Alessio Figalli to the context of stochastic processes. DiPerna and Lions were able to prove the global existence of solutions to the Boltzmann equation. Later, by applying the methods of Fourier integral operators, Lions established