This volume is a collection of original research papers and expository articles stemming from the scientific program of the Nonlinear PDE Emphasis Year held at Northwestern University (Evanston, IL) in March 1998. The book offers a cross-section of the most significant recent advances and current trends and direction in nonlinear partial differential equations and related topics.

The book's contributions offer two perspectives. There are papers on general analytical treatment of the theory and papers on computational methods and applications originating from significant realistic mathematical models of natural phenomena. Also included are articles that bridge the gap between these two perspectives, seeking synergistic links between theory, modeling, and computation. The volume offers direct insight into recent trends in PDEs.

This electronic version is available at no additional charge to purchasers of the print volume. Access instructions are provided in the book. The print volume is also available on the AMS Bookstore: Item code CONM/238. A license agreement is required to purchase the electronic edition only.

Readership

Graduate students and research mathematicians interested in nonlinear partial differential equations and related topics; physicists, engineers, and computational scientists.

Table of Contents

• P. Baiti, P. G. LeFloch, and B. Piccoli -- Nonclassical shocks and the Cauchy problem: General conservation laws
• L. A. Caffarelli -- The Harnack inequality and non-divergence equations
• G.-Q. Chen and H. Frid -- Vanishing viscosity limit for initial-boundary value problems for conservation laws
• A. J. Chorin, A. P. Kast, and R. Kupferman -- On the prediction of large-scale dynamics using unresolved computations
• P. Constantin -- Variational bounds in turbulent convection
• B. Dacorogna and P. Marcellini -- On the solvability of implicit nonlinear systems in the vectorial case
• C. M. Dafermos -- Genuinely nonlinear hyperbolic systems of two conservation laws
• I. M. Gamba -- Milne problem for strong force scaling
• J. Glimm, J. W. Grove, X. L. Li, and N. Zhao -- Simple front tracking
• Y. Guo and A. S. Tahvildar-Zadeh -- Formation of singularities in relativistic fluid dynamics and in spherically symmetric plasma dynamics
• C. Lattanzio and P. Marcati -- Asymptotic stability of plane diffusion waves for the $2$-$D$ quasilinear wave equation
• T.-P. Liu and T. Yang -- $L_1$ stability for systems of hyperbolic conservation laws
• T. Ma and S. Wang -- The geometry of the stream lines of steady states of the Navier-Stokes equations
• R. Magnanini and G. Talenti -- On complex-valued solutions to a 2D eikonal equation. Part one: Qualitative properties
• U. F. Mayer and G. Simonett -- On diffusion-induced grain-boundary motion
• M. O'Leary -- Local estimates for solutions to singular and degenerate quasilinear parabolic equations
• D. Peng, S. Osher, B. Merriman, and H.-K. Zhao -- The geometry of Wulff crystal shapes and its relations with Riemann problems