On incompressible flows in discrete networks and Shnirelman's inequality

Let $f$ and $g$ be two volume-preserving diffeomorphisms on the cube $Q=[0,1]^{\nu}$, $\nu \geq 3$. We show that there is a divergence-free vector field $v \in L^1((0,1);L^p(Q))$ such that $v$ connects $f$ and $g$ through the corresponding flow and $\Vert v \Vert_{L^1_t L^p_x} \leq C_{p,\nu} \Vert f- g \Vert_{L^p_x}$. In particular we show Shnirelman's inequality, cf. [Shnirelman, Generalized fluid flows, their approximation and applications (1994)], for the optimal H\"older exponent $\alpha =1$, thus proving that the metric on the group of volume-preserving diffeomorphisms of $Q$ is equivalent to the $L^2$-distance. To achieve this, we discretise our problem, use some results on flows in discrete networks and then construct a flow in non-discrete space-time out of the discrete solution.