Global solutions to the shallow water system with a method of an additional argument

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 quasi-linear hyperbolic system for a wide class of initial data, we find a sufficient condition on the initial data that guarantee 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 quasi-linear hyperbolic systems in physical rather than characteristic variables.

Keywords:

• shallow water equations
• classical solutions
• method of an additional argument
• global existence

AMS Subject Classifications:

• 35L50
• 35Q35
• 76B03

Acknowledgements

The authors are supported by the Ministry of Education and Science of Russian Federation (the base part of the state task No. 2014/133, project No. 2839).

Notes

No potential conflict of interest was reported by the authors.