ABSTRACT. – We study weakly convergent sequences of suitable weak solutions of heat flows of harmonic maps or approximated harmonic maps. We prove a dimensional stratification for the space-time concentration measure and verify that the concentration measure, viewed as a generalized varifold, moves according to the generalized varifold flow rule which reduces to the Brakke’s flow of varifold provided that the limiting harmonic map flow is suitable. We also establish an energy quantization for the density of the limiting varifold. 2002 Éditions scientifiques et médicales Elsevier SAS AMS classification: 35K55; 58J35

Most wireless communication networks are two-way, where nodes act as both sources and destinations of messages. This allows for “adaptation ” at or “interaction ” between the nodes – a node’s channel inputs may be functions of its message(s) and previously received signals, in contrast to feedback-free one-way channels where inputs are functions of messages only. How to best adapt, or cooperate, is key to two-way communication, rendering it complex and challenging. However, examples exist of channels where adaptation is not beneficial from a capacity perspective; it is known that for the point-to-point two-way modulo 2 adder and Gaussian channels, adaptation does not increase capacity. We ask whether analogous results hold for several multi-user two-way networks. We first consider deterministic two-way channel models: the binary modulo-2 addition channel and a gen-eralization of this, and the linear deterministic channel which models Gaussian channels at high SNR. For these deterministic models we obtain the capacity region for the two-way multiple access/broadcast channel, the two-way Z channel and the two-way interference channel (under certain “partial ” adaptation constraints in some regimes). We permit all nodes to adapt their channel inputs to past outputs (except for portions of the linear high-SNR two-way interference channel where we only permit 2 of the 4 nodes to fully adapt). However, we show that this adaptation is useless from a capacity region perspective. That is, the two-way fully or partially adaptive capacity region consists