Lecture Notes on Time-Dependent PDEs

Jul 30, 2024

Lecture Notes on Time-Dependent PDEs

Lecture Rescheduling

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

Lecture Focus

  • 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.

Topics to be Covered

1. Time-Dependent PDEs

  • 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

2. Existence of Solutions

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

3. Nonlinear Problems

  • 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.

4. Free Boundary Value Problems

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

Energy Methods for Heat Equation

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

Analysis of Heat Equation

  • 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.

Approximation Methods

  • 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.

Convergence and Compactness

  • 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.

Sobolev Spaces and Regularity

  • 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.

Conclusion

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

Upcoming Topic

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