On incompressible flows in discrete networks and Shnirelman’s inequality

In this talk we will show that, given $f$ and $g$ two volume-preserving diffeomorphisms on the cube $Q=[0,1]^\nu$, $\nu\geq 3$, there exists 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 $\|v\|_{L^1_t; L^p_x} \leq C\|f-g\|_{L^p_x}$. In particular we show Shnirelman's inequality, cf. [Shnirelman, Generalized fluid flows, their approximation and applications (1994)], for the optimal Hölder 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 discretize 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. This is a joint work with Stefan Schiffer.