Suppose I have a 1D advection equation in conservation (divergence) form
$\partial_t u(x,t) = -\partial_x [v(x)u(x,t)],$
where $u$ is a conserved quantity in space, and $v$ gives the velocity of the flow of mass in space. I want to know if it is possible to re-write this in a way that describes the rate of change of $u(x,t)$ as a function of incoming mass from all other points in space (let us call these points $y$). In other words, can I write an equation of the form
$\partial_t u(x,t) = \int_y u(y,t) h(y,x) dy,$
where the function $h(y,x)$ gives the proportion of the mass $u(y,t)$ at position $y$ that moves to position $x$ in space in a small unit of time?
I arrived at the advection equation via the divergence theorem, which is why I can write $\partial_t u(x,t)$ in terms of the rate of change in the space at the same point $x$, but intuitively it seems to me there should be another possible formulation like I have described. I suppose the mass is only entering point $x$ from the $y$ points that are already 'very close' to $x$, which is why it is an advective equation rather than a diffusive equation, but I should still be able to sum over all the $y$ points from where the mass is coming from, shouldn't I!?
As I'm a novice in terms of the mathematics and the applications of the advection equation, I would like to know if the question I have asked is well-posed but also whether or not this is a typical problem people think about in applications? Thanks!