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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.
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