On incompressible flows in discrete networks and Shnirelman's inequality

Abstract

Let $f$ and $g$ be two volume-preserving diffeomorphisms on the cube $Q=[0,1{]}^{\nu }$, $\nu \ge 3$. 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 $\Vert v{\Vert }_{L_t^1{L}_x^p}\le C_{p,\nu }\Vert f-g{\Vert }_{L_x^p}$. 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 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.