Consider the simple linear transport equation
with a velocity . Of course the solution is
i.e. the initial datum is just transported in direction of , as the name of the equation suggests. We may also view the solution as not only depending on space and time but also dependent on the velocity , i.e. we write .
Now consider that the velocity is not really known but somehow uncertain (while the initial datum is still known exactly). Hence, it does not make too much sense to look at the exact solution , because the effect of a wrong velocity will get linearly amplified in time. It seems more sensible to assume a distribution of velocities and look at the averaged solutions that correspond to the different velocities . Hence, the quantity to look at would be
Let’s have a closer look at the averaged solution . We write out , perform a change of variables and end up with
In the case of a Gaussian distribution , i.e.
Now we make a time rescaling , denote and see that
So what’s the point of all this? It turns out that the averaged and time rescaled solution of the transport equation indeed solves the heat equation
In other words, velocity averaging and time rescaling turn a transport equation (a hyperbolic PDE) into a diffusion equation (a parabolic PDE).
To end this blog post, consider the slight generalization of the transport equation
where the velocity depends on and . According to Enrique Zuazua it’s open what happens here when you average over velocities…