# Optimal scalar transport using branching pipe flows

# Optimal scalar transport using branching pipe flows

We consider the problem of "wall-to-wall optimal transport," in which we attempt to maximize the transport of a passive temperature field between hot and cold plates. Specifically, we optimize the choice of the divergence-free velocity field in the advection-diffusion equation amongst all velocities satisfying an enstrophy constraint (which can be understood as a constraint on the power required to generate the flow). Previous work established an apriori upper bound on the transport, scaling as the 1/3-power of the flow's enstrophy. Recently, Doering & Tobasco'19 constructed self-similar two-dimensional steady branching flows saturating this bound up to a logarithmic correction to scaling. This logarithmic correction appears to arise due to a topological obstruction inherent to two-dimensional steady branching flows. We present a construction of three-dimensional ``branching pipe flows" that eliminates the possibility of this logarithmic correction and therefore identifies the optimal scaling as a clean 1/3-power law. Also, using an unsteady "branching blob flow" construction, it appears that the 1/3 scaling is, in fact, optimal in two dimensions as well. We discuss the underlying physical mechanism that makes the branching flows "efficient" in transporting heat and present a design of a mechanical apparatus that can transfer heat close to the best possible scenario. We also briefly discuss a few implications to the problem of anomalous dissipation. Finally, inspired by the branching flows, we present a 3D divergence-free autonomous vector field $b \in W^{1, p} \cap C^\alpha$ (for any p < \infty and \alpha < 1) such that the set S of initial conditions for which the trajectories of the associated ODE are not unique is of full Hausdorff dimension, i.e. \dim_{H} S = 3.