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!



Can the conservative form of the advection equation be re-written by replacing the velocity term with an integral over all other points in space?
for $h(y,x)=\nu(y)\partial\_y\delta(x-y)$ one has, upon partial integration:


$$\int\_{-\infty}^\infty u(y,t)h(y,x)\,dy=-\int\_{-\infty}^\infty \delta(x-y)\partial\_y[\nu(y)u(y,t)]\,dy=-\partial\_x [\nu(x)u(x,t)]$$


