# Books written by Pierre-Louis Lions.

Lions received the Fields Medal, for his work on theory of nonlinear partial differential equations, in 1994 while working at the University of Paris-Dauphine. He was the first to give a complete solution to the Boltzmann equation with proof.

## The Master Equation and the Convergence Problem in Mean Field Games

This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity.

## Mathematical Topics in Fluid Mechanics, Volume 1: Incompressible Models

The main emphasis in Volume 1 is on the mathematical analysis of incompressible models. After recalling the fundamental description of Newtonian fluids, an original and self-contained study of both the classical Navier-Stokes equations (including the inhomogenous case) and the Euler equations is given. Known results and many new results about the existence and regularity of solutions are presented with complete proofs. The discussion contains many interesting insights and remarks. The text highlights in particular the use of modern analytical tools and methods and also indicates many open problems.

## Mathematical Topics in Fluid Mechanics, Volume 2: Compressible Models

The second volume is an attempt to achieve a mathematical understanding of compressible Navier-Stokes equations. It is probably the first reference covering the issue of global solutions in the large. It includes unique material on compactness properties of solutions for the Cauchy problem, the existence and regularity of stationary solutions, and the existence of global weak solutions. Written by one of the world's leading researchers in nonlinear partial differential equations, Mathematical Topics in Fluid Mechanics will be an indispensable reference for every serious researcher in the field.

## The Mathematical Theory of Thermodynamic Limits: Thomas-Fermi Type Models

The thermodynamic limit is a mathematical technique for modeling crystals or other macroscopic objects by considering them as infinite periodic arrays of molecules. The technique allows models in solid state physics to be derived directly from models in quantum chemistry. This book presents new results, many previously unpublished, for a large class of models and provides a survey of the mathematics of thermodynamic limit problems. The authors both work closely with Fields Medal-winner Pierre-Louis Lion, and the book will be a valuable tool for applied mathematicians and mathematical physicists studying nonlinear partial differential equations.

## Paris-Princeton Lectures on Mathematical Finance 2004

This is the third volume in the Paris-Princeton Lectures in Financial Mathematics, which publishes, on an annual basis, cutting-edge research in self-contained, expository articles from outstanding specialists, both established and upcoming.

## Paris-Princeton Lectures on Mathematical Finance 2010

This is the fourth volume in the Paris-Princeton Lectures in Financial Mathematics, which publishes, on an annual basis, cutting-edge research in self-contained, expository articles from outstanding specialists, both established and upcoming.

## Multimodal User Interfaces: From Signals to Interaction

The book presents a common theoretical framework for fusion and fission of multimodal information using the most advanced signal processing algorithms constrained by HCI rules, described in detail and integrated in the context of a common distributed software platform for easy and efficient development and usability assessment of multimodal tools.

## On Euler Equations and Statistical Physics

The general goal is to describe mathematically some coherent structures observed in turbulent flows. The first problem we study concerns 2-dimensional flows: we begin with a system of N points vortices interacting with the natural Coulomb-like force and we consider the associated Gibbs measure. We then show how the measure goes, as N goes to infinity, to a stationary measure which is concentrated on very particular stationary solutions of the two-dimensional Euler equations.

## Parabolic Equations with Irregular Data and Related Issues: Applications to Stochastic Differential Equations

This book studies the existence and uniqueness of solutions to parabolic-type equations with irregular coefficients and/or initial conditions. It elaborates on the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations, and also examines the connection between the results on the partial differential equation and the well-posedness of the underlying stochastic/ordinary differential equation.

## Nonequilibrium Problems in Many-Particle Systems

This volume contains the text of four sets of lectures delivered at the third session of the Summer School organized by C.I.M.E. (Centro Internazionale Matematico Estivo). These texts are preceded by an introduction written by C. Cercignani and M. Pulvirenti which summarizes the present status in the area of Nonequilibrium Problems in Many-Particle Systems and tries to put the contents of the different sets of lectures in the right perspective, in order to orient the reader.

## Stochastic Differential Systems, Stochastic Control Theory and Applications

Proceedings of a Workshop, Held at IMA, June 9-19, 1986