## ABSTRACT

Some recent work on the spatial decay of solutions of the Navier-Stokes equations is described here. The motivation for this work is the classic problem of estimation of the entry length for the development of velocity profiles in the laminar flow of a viscous, incompressible fluid in a pipe or channel. The technique used here is that of the well-developed energy stability theory for the Navier-Stokes equations [1], but the first implementation of these techniques on the entry flow problem was done recently by Horgan and Wheeler [2]. (See also [3].) They considered time-independent solutions of the Navier-Stokes equations in a cylindrical channel of arbitrary cross-section and showed that a certain energy functional of the perturbation from the Hagen-Poiseville type flow for this geometry decays exponentially in the axial direction, provided that the Reynold’s number of the base flow does not exceed an explicitly exhibited quantity. In an independent effort the author and V. G. Sigillito [4] considered time-dependent solutions of the Navier-Stokes equations in a channel which need not be cylindrical. In that work a functional of the flow which depends on the inlet distance and time is shown to decay exponentially with the inlet distance. The details of this work together with some recent extensions will be described in what follows.