Homogenization of Porous Medium Equation

This is a joint project between

Adam Oberman Main Page
Inwon Kim Inwon's Home Page at the University of California, Los Angeles

For background, see Notes on Porous Medium Equation

The goal of this project is to understand how small scale velocity fields average out (homogenize) in the porous medium equation. Away from density zero, we expect the solution to be averaging. However, at density zero, the diffusion coefficient of the equation is zero, so it's possible that the velocity doesn't average out.

To investigate this, we perform numerical simulations in one dimension, of the simplest case of the equation.

$u_t = \partial_{xx} (u)^2/2 + \partial_{x} (v u)$

Here $v = ε w (x / ε)$

We expect that the free boundary velocity is

$V n = | u x | - v x$

Heuristics for the free boundary velocity

Heuristics behind this are as follows. Near the point where $u = 0$ the equation

$u_t = u u_{xx} + |u_x|^2 + \partial_{x} (v u)$

looks like

$u t = | u x | 2 + v u x$

so the normal velocity is

$u_t/|u_x| = |u_x| \pm v$

Pinning by a small scale velocity field

The last equation suggests pinning occurs when the gradient of u is equal to the velocity. I.e.

$P = {u x = v}$ where P is the pinning set.

The next conjecture, which we verify numerically, is that if we take the velocity field

$v = V 0 + A 0 sin(x / ε)$

i.e., if we add a small scale oscillatory velocity field to the constant velocity field, then in the limit that the scale goes to zero, the pinning set is

$P = \{ u_x \in [V_0 - A_0, V_0+ A_0 ] \}$

Homogenization of Porous Medium Equation - Adam Oberman Math

Contents

Introduction

Heuristics for the free boundary velocity

Pinning by a small scale velocity field

HPME old 1

PME Computations, No velocity field

HPME Computations, pinning by a constant velocity field

HMPE Computations, Pinning on both sides

Pinning by constant plus small scale velocity field

HPME More Pinning