This volume is a collection of refereed original research papers and expository articles
and stems from the scientific program of the 1997-98 Nonlinear PDE Emphasis Year at
Northwestern University, which was jointly sponsored by Northwestern University and the
National Science Foundation. Most of the papers presented are from the distinguished
mathematicians who spoke at the International Conference on Nonlinear Partial Differential
Equations, March 21-24, 1998, Evanston, IL.The book is a cross-section of the most
significant recent advances and current trends and directions in nonlinear partial
differential equations and related topics. Contributions range from modern approaches to
the classical theory in elliptic and parabolic equations to nonlinear hyperbolic systems
of conservation laws and their numerical treatment.
The general guiding idea in editing this volume has been twofold. On one hand, we have
solicited the papers that contribute in a substantial way to the general analytical
treatment of the theory of nonlinear partial differential equations. On the other hand, we
have attempted to collect the contributions to computational methods and applications,
originating from significant realistic mathematical models of natural phenomena, to seek
synergistic links between theory and modeling and computation and to underscore current
research trends in partial differential equations. The borderline between these two
aspects of mathematical research is rather fuzzy. We have also selected a set of papers
that would bridge them.
Examples of the first kind of contributions include new insights into the role of the
Harnack inequality in the theory of fully nonlinear elliptic equations, new results on the
local behavior of degenerate parabolic equations, a treatment of the complex eikonal
equations, and the solvability of implicit degenerate elliptic systems and motion by
curvature.
Included in this broad category also are the papers establishing the regularity,
large-time behavior, and L1 stability of entropy solutions, the analysis
of non-classical shocks, and the convergence of the vanishing viscosity method for
initial-boundary value problems for nonlinear hyperbolic systems of conservation laws, as
well as the asymptotic stability of diffusion waves for the multidimensional nonlinear
wave equations and the structural stability of steady-state solutions of the Navier-Stokes
equations.
Contributions of the second kind range from numerical methods for predicting the
large-scale dynamics and multidimensional simple front tracking algorithms to mathematical
aspects of turbulent convection, geometry of crystal shapes, singularities in relativity
and plasma dynamics, and high field kinetic semiconductor models.
This volume would not have been possible without the help and support of a number of
people and institutions. First, we would like to thank the American Mathematical Society,
especially, Edward G. Dunne (Editor of Book Program), Christine M. Thivierge (Acquisitions
Assistant), Deborah Smith (Production Editor), and the technical support group for their
prompt and professional assistance and their patience with our slow pace.
Karen Townsend deserves our special thanks for her assistance in the review process.
We are also grateful to the referees for their constructive criticisms and suggestions,
and to Konstantina Trivisa and Mikhail Feldman for their invaluable help with the
editorial work.
Finally, we wish to acknowledge the financial support of the National Science
Foundation through grant DMS-9708261 and Northwestern University, more specially, the
Department of Mathematics and the Office of the Vice President for Research of
Northwestern University.
The purchaser of this volume is entitled to the online version of this book by the AMS.
To gain access, follow the instructions given on the form found in the back of this
volume.
Gui-Qiang Chen and Emmanuele DiBenedetto
Evanston, Illinois
