Existence of Global Classical Solutions for the Saint-Venant Equations

Nowadays, investigations of the existence of global classical solutions for non linear evolution equations is a topic of active mathematical research. In this article, we are concerned with a classical system of shallow water equations which describes long surface waves in a fluid of variable depth. This system was proposed in 1871 by Adh\'{e}mar Jean-Claude Barr\'{e} de Saint-Venant. Namely, we investigate an initial value problem for the one dimensional Saint-Venant equations. We are especially interested in question of what sufficient conditions the initial data and the topography of the bottom must verify in~order that the considered system has global classical solutions. In order to prove our main results we use a~new topological approach based on the fixed point abstract theory of the sum of two operators in Banach spaces. This basic and new idea yields global existence theorems for many of the interesting equations of mathematical physics.