Homogenisation Theory for PDEs Homogenisation for Advection-Diffusion Equations

Starting point Homogenisation Homogenised equation ?

Starting point Review- Steady state heat conduction Goal: Homogenised equation several assumptions…

macroscopic scale microscopic scale Independent Variables Multiple Scale Method Assume the expansion Validity?

Solvability condition for : By inserting the expansion into, we obtain

Instead of, we now have Homogenised Equation Cell problem:assume that, where

Starting point Steady state heat conduction Homogenised equation

There exists a unique solution of Existence, uniqueness and convergence and There exists a unique solution of weakly in.

Remark (validity of the expansion) cell problem higher order cell problem, solution of the homogenised problem Under certain conditions, we get the estimate higher order cell problems

Advection-Diffusion equations incompressible: given, 1-periodic and sufficiently smooth passive tracer

Linear Transport equations ( ) Where is ?

New variables Goal: Rescaling New formulation of the problem

Multiple Scale Method By substituting the expansion in the equation, we obtain Problem!!! Example:where

If Then indeed By computing we obtain the homogenised equation, where ergodic

Advection-Diffusion equations ( ) incompressible: Given, 1-periodic and sufficiently smooth passive tracer

New variables Rescaling New formulation of the problem Goal:

By substituting the expansion in the equation, we obtain Multiple Scale Method where

Solvability Condition Integrate over Y Solvability condition smooth 1-periodic function

First step ( ) In fact,

Solvability condition Separation of variables Using this in, we obtain the cell problem Second step ( )

Solvability condition Leads to the homogenised equation Effective Diffusivity: Third step ( ) where matrix

Effective Diffusivity For every vector we have The homogenisation procedure enhances diffusion; the effective diffusivity is always greater than the molecular diffusivity in the following sense:

Summary Steady state heat conduction (Review) Multiple Scale Method Existence, uniqueness and convergence Remark (validity of the expansion) Advection-Diffusion equations (linear transport equation)