A convection±diusion problem in a network 1 Seth F. Oppenheimer 2 Department of Mathematics and Statistics, Mississippi State University, P.O. Drawer MA, Mississippi State, MS 39759, USA Abstract In recent years, it has become increasingly clear that the standard method of mod- eling hydrodynamic dispersion in groundwater ¯ow by diusion is often unsatisfactory. In the last few years, several authors have tried to overcome this problem by modeling subsurface ¯ow by ¯ow in a network. This has had some success and even some agreement with experiment. This work will lay the mathematical foundations for such models which, up to this point in time, have been studied only numerically. A well- posedness result is obtained for a class of these models. Some qualitative properties of solutions, such as positivity, ordering, and boundedness are also given. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Networks; Convection; Diusion; Flow in porous media 1. Introduction In recent years, it has become increasingly clear that the standard method of modeling hydrodynamic dispersion in groundwater ¯ow by diusion is often unsatisfactory. An excellent description of the history of this problem may be found in [5]. In the last few years, several authors have tried to overcome this problem by modeling subsurface ¯ow by ¯ow in a network (see [2,3,5,7,8,10], and [11]). The goal is not to obtain an exact description of the actual Applied Mathematics and Computation 112 (2000) 223±240 www.elsevier.nl/locate/amc 1 This work was partially supported by US Army contract DACA39±94±0018. 2 E-mail: seth@math.msstate.edu 0096-3003/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII:S0096-3003(99)00045-4