Turning transport into diffusion

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

regularize

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\$ ofâ€¦

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