Global solutions to the shallow-water system

TL;DR

This paper establishes conditions under which the shallow-water equations have global classical solutions without shock formation, using a novel method in physical variables to analyze the system's long-term behavior.

Contribution

It introduces a new approach to prove the existence of global solutions for the shallow-water system under specific initial conditions, avoiding shock formation.

Findings

Global classical solutions exist under certain initial data conditions.

Solutions maintain monotonic Riemann invariants over time.

No shock wave singularities form in finite time under these conditions.

Abstract

The classical system of shallow-water (Saint--Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasilinear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantees existence of a global classical solution continued from a local solution. The sufficient conditions can be easily satisfied for the fluid flow propagating in one direction with two characteristic velocities of the same sign and two monotonically increasing Riemann invariants. We prove that these properties persist in the time evolution of the classical solutions to the shallow-water equations and provide no shock wave singularities formed in a finite time over a half-line or an infinite line. On a technical side, we develop a novel method of an additional argument, which allows to obtain local and global solutions to the quasilinear hyperbolic systems in physical rather than characteristic variables.