Jul 30, 2024

- The lecture is rescheduled from its original date due to absence from earlier sessions.
- New start time:
**8:30 AM on Thursdays**.

- The module description is deemed ambitious and misaligned with the students' current level.
- The focus will shift to more basic concepts covering research topics for advanced semestering.

- We'll focus on second-order equations in space, notably:
- Navier-Stokes Equation
- Stokes Equation
- Heat Equation
- Diffusion Equation

- Examination will also include first-order equations like:
- Euler Equations
- Wave Equations

- Initial focus on procedures to demonstrate existence for linear PDE solutions.
- Understanding of linear problems simplifies analysis.

- Future sessions will address nonlinear PDEs, focusing on:
- Non-linear elasticity and deriving models related to hyperelastic materials.
- Introduction to calculus of variations and energy methods.

- Exploring free boundary value problems like phase transition modeling where part of the boundary condition is unknown.

- Starting with
**Heat Equation**:- Notation: ( u_t - \Delta u = 0 )
- Boundary conditions are kept homogeneous, and an inhomogeneous initial condition will be required.

- Objective: To generalize solutions beyond the basic heat equation to more complex operators.
- Discuss separation of variables and eigenfunctions:
- Linear ordinary differential equations arising from approximations of the heat equation.
- Interpretation of initial conditions through Fourier coefficients for approximation.

- Distinction between discretizing in time versus space.
- Implicit Euler Method serves as a basis for numerical analysis.
- Setting up systems of elliptic boundary problems based on discretized time settings.

- Importance of weak convergence in infinite dimensional spaces versus lower dimensions, especially for bounded sequences.
- Utilize
**Azela-Ascoli theorem**to assess compact sets in function spaces.

- Explore the concept of Sobolev spaces in various contexts:
- Define and examine properties of these spaces in relation to the trace properties.
- Use of continuous functions as a baseline for analysis.

- Aim for understanding the finite and infinite dimensional dynamics within PDEs.
- Expect to derive a priori estimates and convergence conclusions for future lectures.

- Continue with the implicit Euler Method next week, focusing on deriving estimates and existence proof details.