Title
Applications of branching flows to optimal scalar transport and the nonuniqueness of trajectories for Sobolev vector fields
Abstract
We are interested in the design of forcing in the Navier–Stokes equation such that the resultant flow maximizes the transport of a passive temperature between two differentially heated walls for a given power supply budget. Previous work established that the transport cannot scale faster than 1/3-power of the power supply. Recently, Doering & Tobasco (CPAM’19) constructed self-similar two-dimensional steady branching flows, saturating this upper bound up to a logarithmic correction to scaling. We present a construction of three-dimensional “branching pipe flows” that eliminates the possibility of this logarithmic correction and for which the corresponding passive scalar transport scales as a clean 1/3-power law in power supply. However, using an unsteady branching flow construction, it appears that the 1/3 scaling is also optimal in two dimensions. After carefully examining these designs, we extract the underlying physical mechanism that makes the branching flows “efficient,” based on which we present a design of mechanical apparatus that, in principle, can achieve the best possible case scenario of heat transfer.
Finally, we present an application of branching flows to a study concerning the nonuniqueness of trajectories. After the theory of DiPerna–Lions’89, a question remained, whether there are continuous Sobolev vector fields such that the trajectories are not unique on a set of positive measures. Recently, an answer to this question was given, see, for example, Bruè, Colombo and De Lellis’21 and Fefferman et al.’21. Borrowing a few ideas from our study on optimal scalar transport, we present an explicit construction of a vector field that fully resolves this question.
Title (Jakub Skrzeczkowski)
TBC
Abstract (Jakub Skrzeczkowski)
TBC
Please note that the seminar will take place in person in room 140 of Huxley Building.