Pinning by constant plus small scale velocity field
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Example from Homogenization of Porous Medium Equation
The next conjecture, which we verify numerically, is that if we take the velocity field
v = V0 + A0sin(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 ] \}](/oberman/w/images/math/0/1/7/0178fb504a04b917c2c464115d1d275d.png)
where the derivative is taken at the free boundary (actually we start with a linear profile for the function).
The following computational examples show the steady state, pinned front, which occurs after 2-4 units of time.
Pinning with slope 1, velocity 1 plus oscillating velocity field
The next example shows pinning by a constant velocity field (the constant is 1) plus a small scale oscillatory velocity field. Here the pinning would occur even without the small scale velocity field.
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Pinning with slope .95, velocity 1 plus oscillating velocity field
The next example shows pinning by a constant velocity field (the constant is 1) plus a small scale oscillatory velocity field. Here the pinning would fail without the small scale velocity field.
We are seeing 5 snapshots, over a time interval of 1, so the speed (without the oscillations) would be .05, and the distance would be .05.
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Pinning with slope 1.05, velocity 1 plus oscillating velocity field
The next example shows pinning by a constant velocity field (the constant is 1) plus a small scale oscillatory velocity field. Here the pinning would fail without the small scale velocity field.
We are seeing 5 snapshots, over a time interval of 1, so the speed (without the oscillations) would be .05, and the distance would be .05.
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Pinning
The next example shows pinning by a constant velocity field (the constant is 1) plus a small scale oscillatory velocity field. Here the pinning would occur even without the small scale velocity field.
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Example from Homogenization of Porous Medium Equation
Pinning with slope 1, velocity .9 plus oscillating velocity field
The next example shows pinning by a constant velocity field (the constant is 1) plus a small scale oscillatory velocity field. Here the pinning would fail without the small scale velocity field.
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Pinning with slope 1, velocity 1.1 plus oscillating velocity field
The next example shows pinning by a constant velocity field (the constant is 1) plus a small scale oscillatory velocity field. Here the pinning would fail without the small scale velocity field.
We are seeing 5 snapshots, over a time interval of 1, so the speed (without the oscillations) would be .05, and the distance would be .05.
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