Global Existence and Stability of Solutions to River Flow System

Abstract

In this short note, we are concerned with the global existence and stability of solutions to the river flow system. We introduce a new technique to set up a relation between the Riemann invariants and the finite mass to obtain a time-independent, bounded solution for any adiabatic exponent. The global existence of solutions was known long ago [Klingenberg and Lu in Commun. Math. Phys. 187: 327–340, 1997]. However, since the uncertainty of the function b(x), which corresponds physically to the slope of the topography, the $L^{\infty }$ estimates growed larger with respect to the time variable. As a result, it does not guarantee the stability of solutions. By employing a suitable mathematical transformation to control the slope of the topography by the friction and the finite mass, we prove the uniformly bounded estimate with respect to the time variable. This means that our solutions are stable.