With a simple physical motivation, we will learn several fundamental linear equations that form the basis of partial differential equations (PDEs). After studying the characteristic method for ordinary differential equations (ODEs) and conservation laws, the latter part of the lecture will cover key aspects of modern PDE theory, such as Sobolev spaces, traces, Sobolev embedding theorems, compactness results, and more. These concepts will then be applied to study the existence, uniqueness, and regularity estimates of solutions to linear evolution equations.

- Basic undergraduate courses related to analysis
- Graduate-level real analysis (including measure theory)
- It's even better if you have studied functional analysis, but not required

Midterm exam 50%, final exam 50%

1.J. Smoller, Shock Waves and Reaction Diffusion Equations, Springer (1994)
2. G. B. Whitham, Linear and Nonlinear Waves,Wiley and Sons (1974).
3. G. I. Barenblatt. Scaling, self-similarity and intermediate asymptotics. Cambridge
University Press, Cambridge, 2 edition, 1996.

I. Nonlinear first order partial differential equations
1.1. Characteristics
1.2. Hamilton-Jacobi equations (Weak solutions)
1.3. Conservation laws
1.4. system of conservation laws

II. Some ways to represent solutions in closed forms
2.1. Self similarity in boundary value problems
2.2. Self-similarity of second kind

III. Reaction-diffusion systems
3.1. Local existence of solutions
3.2. Invariant regions (Turing instability and regularity for parabolic equations

In-person lectures.

Lectures will be uploaded online during business trips.