Problems Solved and Unsolved Concerning Linear and Nonlinear Partial Differential Equations

by Peter D. Lax

A concise survey of major advances in linear and nonlinear partial differential equations, with practical hints for future work and applications.

This edition introduces broad topics in PDE theory, then surveys two focused linear problems and nonlinear problems. It highlights how modern tools—such as microlocal analysis, trace formulas, and spectral theory—have reshaped our understanding of wave propagation, diffraction, and the behavior of complex systems. The discussion ranges from geometric settings on manifolds to concrete problems in fluid dynamics and conservation laws, offering a snapshot of how ideas from analysis, geometry, and computation come together in PDE research.

- Learn how linear PDEs on manifolds with boundaries have seen significant progress, including wave propagation, diffractive phenomena, and spectral questions.
- See how nonlinear PDEs are approached, from viscous incompressible flows to hyperbolic conservation laws and completely integrable systems.
- Discover the interplay between energy methods, weak solutions, and modern techniques like numerical experimentation and translation representations.
- Get a sense of how non-Euclidean geometry and spectral theory influence PDE behavior, including the Laplace-Beltrami operator on negatively curved spaces.

Ideal for readers of advanced mathematics and mathematical physics who want a clear, broad view of current challenges and methods in PDE research.

Kindle Edition

First published August 13, 2015

About the author

Peter David Lax was a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics.
Lax has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields.
In a 1958 paper Lax stated a conjecture about matrix representations for third order hyperbolic polynomials which remained unproven for over four decades. Interest in the "Lax conjecture" grew as mathematicians working in several different areas recognized the importance of its implications in their field, until it was finally proven to be true in 2003.