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I have the spatial density u(x,t) of a substance, and I want to describe the simple transport of this substance along a given vector field phi(x,t). Am I correct that the corresponding equation is

$\frac{\partial}{\partial t} u(x,t) = \phi(x,t)\cdot\nabla_x u(x,t)$?

I am confused because someone (possibly unrelyable) told me it was

$\frac{\partial}{\partial t} u(x,t) = \nabla_x \cdot (\phi(x,t)\cdot u(x,t))$?

When i google "transport equation", $\phi$ is always a constant, for which the two equations coincide..

If it's the first equation: Using the method of characteristics, I get $\frac{d x(t)}{dt}=\phi(x(t),t)$ for the characteristic curves, an equation I cannot solve. Hence there is no way for me to solve the PDE, right? Thanks.

kamui
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1 Answers1

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It's the second one, but you have the sign reversed. Correct form is $$u_t + \nabla_x \cdot (u \phi) =0 \tag{1}$$ where $\phi$ is velocity and $u$ is concentration of substance. Reason: $u\phi$ represents the mass flow rate, and taking its divergence gives the rate at which the mass escapes from a neighborhood of $x$. This is the rate at which the concentration is dropping at that point.

Note that (1) can be written as divergence of one space-time field: $$ \nabla_{x,t} \cdot (u\phi, u) = 0 $$ which is the law of concentration of mass written in space-time coordinates.