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This is mostly a reference request question, although I certainly appreciate any insights and/or comments.

Let us assume $p:R^n×(0,∞)\to \mathbb R$ is a scalar concentration, $u\in R^n$ is the underlying continuous velocity field, $x\in R^n$ are the physical coordinates, t is time.

A typical advection (or transport) equation in absence of diffusion is given by:

$$\frac{\partial p(x,t)}{\partial t}+\nabla\cdot(p(x,t)u(x))=0$$

Further assume velocity field is divergence free. In some circumstances (i.e. when the scalar is passive), the underlying velocity field is not affected by the flow of the scalar. Hence, here $u(x)$ is constant, and the scalar trajectories correspond with the streamlines of the velocity field, which are given as solution of:

$$\frac{dx}{dt}=u(x)$$

Now if we add diffusion, we get: $$\dfrac{\partial p(x,t)}{\partial t}+\nabla\cdot(p(x,t)u(x))=K\nabla^2p(x,t)$$

But this can be written as :

$$\frac{\partial p(x,t)}{\partial t}+\nabla\cdot\left [ p(x,t)\left [u(x)-K\frac{\nabla p(x,t)}{p(x,t)}\right ] \right ]=0 $$

we have made the "effective" velocity field $v$ dependent on scalar $p$.

where $v(x,t)=u(x)-K\dfrac{\nabla p(x,t)}{p(x,t)}$ plays the role of the new 'velocity' field.

Now $p(x,t)$ represents the scalar transport (advection without diffusion) under this modified velocity field.

My question is if this approach has been explored in studying the original advection-diffusion equations in the literature. I feel there is additional insight to be gained by this change of viewpoint, although I don't have concrete proof of that.

What I mean is that now we can study the dynamical system:

$$\frac{dx}{dt}=v(x),$$ and apply techniques from dynamical systems to study fixed points etc. Obviously, the dependence of $v$ on $p$ complicates things here, but the approach seems to me as something useful to pursue.

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