The articles of this book are written by leading experts in partial differential equations and their applications, who present overviews here of recent advances in this broad area of mathematics. The formation of shocks in fluids, modern numerical computation of turbulence, the breaking of the Einstein equations in a vacuum, the dynamics of defects in crystals, effects due to entropy in hyperbolic conservation laws, the Navier-Stokes and other limits of the Boltzmann equation, occupancy times for Brownian motion in a two dimensional wedge, and new methods of analyzing and solving integrable systems are some of this volume's subjects. The reader will find an exposition of important advances without a lot of technicalities and with an emphasis on the basic ideas of this field.

Readership

Graduate students and research mathematicians interested in partial differential equations.

Table of Contents

• S. Klainerman -- Null hypersurfaces with finite curvature flux and a breakdown criterion in general relativity
• D. Christodoulou -- The formation of shocks in 3-dimensional fluids
• F. A. Grünbaum and C. McGrouther -- Occupation time for two dimensional Brownian motion in a wedge
• R. Buckingham, A. Tovbis, S. Venakides, and X. Zhou -- The semiclassical focusing nonlinear Schrödinger equation
• Y. Li -- An extension to a classical theorem of Liouville and applications
• A. S. Fokas -- From Green to Lax via Fourier
• J. Jimenéz -- Untangling wall turbulence through direct simulations
• L. L. Bonilla and A. Carpio -- Defects, singularities and waves
• C. D. Levermore -- Fluid dynamics from Boltzmann equations
• F. Golse -- From the Boltzmann equation to the incompressible Navier-Stokes equations
• C. M. Dafermos -- Hyperbolic conservation laws with involutions and contingent entropies