Lecture Series: Theory of Partial Differential Equations
Introduction
- Unique theoretical perspective on partial differential equations (PDEs).
- Unlike typical applied approaches found on platforms like YouTube.
- Course will cover theoretical aspects mostly overlooked.
- First video focuses on syllabus/content overview; subsequent videos will dive deeper.
Course Logistics
- Frequency: 2-3 lectures per week (could be more).
- Homework: Assignments provided after 4-5 lectures; solutions will be submitted and reviewed.
- Resources: Link to GitHub page for syllabus and lectures.
Prerequisites
- Advanced calculus
- Advanced mathematical analysis
- Functional analysis
- Measure Theory
- Note: Basic real analysis is generally sufficient.
Course Content Overview
Revision Topics
- Functional Analysis: Banach and Hilbert spaces, dual spaces, Riez Representation Theorem, Lax-Milgram Theorem, weak and weak-star convergence.
- Measure Theory: Lebesgue measure and integral.
Function Spaces
- LP spaces
- Convolutions and mollifiers
- Functional inequalities (important in PDE analysis)
Sobolev Spaces and Weak Derivatives
- Sobolev spaces and calculus
- Poincaré inequality and embedding theorems
Linear Elliptic Equations
- Existence and uniqueness of solutions
- Regularity and elliptic regularity (e.g., Laplace’s equation)
- Hormander’s function expansion
Fourier Series and Galerkin Method
- Introduction to heat equation
- Weak solutions for linear parabolic equations
Advanced Topics
- Space of distributions
- Fundamental solutions, Fourier transform, and L2 Hilbert spaces
Advanced PDE Topics (Optional)
- Existence and uniqueness for Navier-Stokes equations (2D and 3D)
- Recent studies and papers on Navier-Stokes (2D focus for practical discussions)
- Reference to work by Jean Leray (1930s) and modern interpretations. Optional deep dive into Leray’s paper in French and its modern counterpart.
Recommended Textbooks
- Partial Differential Equations by Evans (widely used, available in market, Amazon, eBay).
- Introduction to Partial Differential Equations by David Borthwick (good mix of pure and applied perspectives).
- Infinite Dimensional Dynamical Systems by James C. Robinson (focus on existence and uniqueness, Galerkin method).
Next Steps
- First lecture: Functional analysis and major results.
- Second lecture: Measure theory and key results.
- Future lectures: Deep dive into partial differential equations.
Stay tuned for the first lecture!