There are many papers devoted to the study of shallow water flows especially by numerical methods. In this note, we give a mathematical analysis, based on the theory of semigroups of operators on Hilbert space, of two linearized problems involving the Saint-Venant equations governing shallow water flows. The first one concerns the classical linearized Saint-Venant problem, while the second one deals with the water quality problem: the evolution of a concentration of products in a flow solution of the first problem.
"On some linearized problems of shallow water flows." Differential Integral Equations 22 (3/4) 275 - 283, March/April 2009.