Applied Partial Differential Equations

Outline

Applied PDEs play a dual role in applied math. It is part of the subject matter of applied math. For example, mathematical theory of nonlinear conservation laws and asymptotic analysis of wave propagation have been very important topics in applied math. It provides the background and techniques for other areas of the applied math. (Example: Numerical techniques for conservation laws have drawn inspiration from the theory of Lax, Kruskov, Volpert, etc.)

About selection of topics:

1. Basic theory of differential equations, e.g. well-posedness, qualitative behavior, properties of basic PDEs.
2. Basic techniques: e.g. Fourier transforms, asymptotics, method of characteristics, scaling.
3. Examples of PDEs that are often encountered in applied math: e.g. Euler equations, Ginzburg-Landau equations, Burgers equation, Hamilton-Jacobi equations.
4. Culture: What an educated applied mathematician needs to know about differential equations, e.g. KAM theorems.

About the style:

Seek to present the subject matter in the most straightforward fashion. Combination of rigorous analysis, heuristic arguments that provides insight. Always places insight above formality.

Comment on the contents:

The first half is on basic theory and techniques. The second half is on selected examples that are often encountered in applied math.