Turning transport into diffusion

A remark very interesting where transport equations is linked to heat equations.


Consider the simple linear transport equation

$latex displaystyle partial_t f + vpartial_x f = 0,quad f(x,0) = phi(x) &fg=000000$

with a velocity $latex {v}&fg=000000$. Of course the solution is

$latex displaystyle f(x,t) = phi(x-tv), &fg=000000$

i.e. the initial datum is just transported in direction of $latex {v}&fg=000000$, as the name of the equation suggests. We may also view the solution $latex {f}&fg=000000$ as not only depending on space $latex {x}&fg=000000$ and time $latex {t}&fg=000000$ but also dependent on the velocity $latex {v}&fg=000000$, i.e. we write $latex {f(x,t,v) =phi(x-tv)}&fg=000000$.

Now consider that the velocity is not really known but somehow uncertain (while the initial datum $latex {phi}&fg=000000$ is still known exactly). Hence, it does not make too much sense to look at the exact solution $latex {f}&fg=000000$, because the effect of a wrong velocity will get linearly amplified in time. It seems more sensible to assume a distribution $latex {rho}&fg=000000$ of…

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