Continuous Maximum Flow Problem Maximum Flow Problem, but continuous.
Suppose there are two points on a plane, the $source$ and the $sink$. There's also $capacity$ of each point on the plane: the maximum amount that can flow through this point in a unit of time. Capacity is a continuous function $\left( \Bbb R , \Bbb R \right) \to \Bbb R_{\ge 0}$.
A $flow$ is a series of $streams$. A stream is a pair: $rate \ of \ flow$ ($\Bbb R_{\ge 0}$), the amount that actually flows in a unit of time, plus the $path of flow$, the points that the flow goes through, also continuous: $$\left[ 0, 1 \right] \to \left( \Bbb R , \Bbb R \right), 0 \to source, 1 \to sink$$
A flow is $invalid$ if there's a point such that the sum of all rates of flow through it is higher than it's capacity.
Is it possible to calculate the maximum valid flow and construct an example of such a flow?
Bonus: prove an equivalent of the Max-flow min-cut theorem and construct an example of a cut!
 A: This paper gives a treatment of the general treatment of the continuous min-flow/max-cut duality. I won't go into the details of the proof of duality, but the general setup of the problem is as follows:
In place of a network, let $\Omega\subset\mathbb{R}^n$ be a closed, bounded subset whose boundary $\Gamma$ is a smooth surface with unit normal $n$. A flow field $\sigma:\Omega\to\mathbb{R}^n$ is a smooth vector field. The sources and sinks can be distributed over the interior and boundary by two functions $F:\Omega\to\mathbb{R}$ and $f:\Gamma\to\mathbb{R}$ which determine the relative strength of sources, which impose the folowing constraints, where $\lambda$ allows al in/outflows to be scaled together.:
$$
\text{div}\ \sigma=\lambda F,\ \ \ n\cdot\text{div}\ \sigma=\lambda f
$$
Additionally, there is a maximum flow rate at each point, given by a smooth function $c:\Omega\to R$, which imposes the constraint $\|\sigma\|\le c$. The overall problem can be stated as
$$
\text{maximize}_{\sigma,\lambda}\ \ \ \lambda \\
\text{subject to}\ \ \ \text{div}\ \sigma=\lambda F,\ \ \ n\cdot\text{div}\ \sigma=\lambda f,\ \ \ \|\sigma\|\le c
$$
Your description of the problem uses two points, a source and sink, which I'll call $p$ and $q$ respectively. We can fit this into the above language by allowing $\sigma$ to be undefined at $p$ and $q$ and letting $F(x)=\delta(x-p)-\delta(x-q)$ and $f(x)=0$, where $\delta$ is the Dirac delta function.
If you want to use all of $\mathbb{R}^n$, there is a problem of sources and sinks "at infinity". These can be eliminated by requiring $\lim_{r\to\infty} \int_{S(r)}c=0$, where $S(r)$ is a sphere of radius $r$ centered at the origin. The arguments in the paper may require additional care to verify that integrals converge, but the same duality should apply.
