This course introduces several fundamental linear equations that arise in mathematical physics and form the foundation of modern partial differential equations (PDEs). We begin with the theory of distributions and weak derivatives, followed by essential topics in ordinary differential equations, including existence and uniqueness theory, Grönwall's inequality, bootstrap arguments, and conservation laws. After developing the method of characteristics for first-order equations, we study core tools of modern PDE analysis, such as Sobolev spaces, trace operators, Sobolev embedding theorems, and compactness methods. These analytical techniques are then applied to investigate the existence, uniqueness, and regularity properties of solutions to classical 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

G.B. Folland, Real Analysis: Modern Techniques and Their Applications

Week 1. Introduction and Physical Motivation
* Examples from physics:
* Transport equation
* Heat equation
* Wave equation
* Classification of PDEs
* Initial and boundary value problems
* Well-posedness (Hadamard)

Week 2. Theory of Distributions I
* Test functions
* Distributions
* Weak derivatives
* Examples:
* Dirac delta
* Heaviside function
* Principal value distributions

Week 3. Theory of Distributions II
* Operations on distributions
* Differentiation
* Multiplication by smooth functions
* Convolution
* Fundamental solutions

Week 4. ODE Theory I
* Local existence and uniqueness
* Picard iteration
* Lipschitz conditions
* Continuation criteria

Week 5. ODE Theory II
* Grönwall inequalities
* Continuous dependence on initial data
* Stability
* Bootstrap arguments

Week 6. ODE Theory III
* Flows generated by vector fields
* Conserved quantities
* Noether-type principles
* Applications to dynamical systems

Week 7. Method of Characteristics I
* Linear transport equations
* Characteristic curves
* Explicit solution formulas
* Propagation of regularity

Week 8. Method of Characteristics II
* First-order quasilinear equations
* Burgers equation
* Shock formation
* Conservation laws

Week 9. Conservation Laws
* Weak solutions
* Entropy condition
* Rankine–Hugoniot condition
* Finite speed of propagation

Week 10. Sobolev Spaces I
* (L^p) spaces review
* Weak derivatives
* Definition of (W^{k,p})
* Basic properties

Week 11. Sobolev Spaces II
* Mollifiers
* Density theorems
* Approximation by smooth functions
* Poincaré inequality

Week 12. Sobolev Embeddings and Traces
* Sobolev embedding theorem
* Morrey inequality
* Trace theorem
* Boundary values

Week 13. Compactness Methods
* Arzelà–Ascoli theorem
* Rellich–Kondrachov compactness theorem
* Weak convergence
* Compactness techniques in PDE

Week 14. Linear Evolution Equations I
* Heat equation
* Energy estimates
* Maximum principle
* Existence and uniqueness

Week 15. Linear Evolution Equations II
* Wave equation
* Duhamel principle
* Regularity estimates
* Course summary and outlook toward nonlinear PDEs

In-person lectures.

Lectures will be uploaded online during business trips.