Why is the equation $u_t + (a u)_x = 0$ conservative but $u_t + a u_x = 0$ isn't? Consider the following advection equations:
$$u_t + a u_x = 0$$
$$u_t + (au)_x = 0$$
I have heard the second one described as being conservative, yet the first one isn't. Why is this?

Physically speaking, ''conservative'' means that the total mass of $u$ is conserved in time. To see this, suppose $u : [0,T] \times [0,1] \to \mathbb R$ with $u(t,0) = u(t,1)=0$ for all $t \in [0,T]$ and define the mass of $u$ at time $t$ by
$$M(t)= \int_0^1 u(t,x)dx.$$
If $u$ satisfies the second (conservative) equation, you get
$$ M'(t) = \int_0^1 \partial_t u(t,x)dx = -\int_0^1 \partial_x(a(t,x)u(t,x))dx = u(t,0)a(t,0)-u(t,1)a(t,1) = 0$$
Therefore the mass of $u$ is constant in time, so $u$ is conserved. In the first equation things go wrong because you get, after an integration by parts
$$M'(t) = -\int_0^1 a(t,x) \partial_x u(t,x)dx = \int_0^1 u(t,x) \partial_x a(t,x)dx + u(t,0)a(t,0)-u(t,1)a(t,1)$$
$$= \int_0^1 u(t,x) \partial_x a(t,x)dx.$$
and so $M$ is in general not constant in time: equation non conservative.
