A remark on the Glimm scheme for inhomogeneous hyperbolic systems of balance laws

TL;DR

This paper proves the global existence of entropy weak solutions for inhomogeneous hyperbolic balance laws with small BV data, without assuming equilibrium or specific source dependence, extending previous results.

Contribution

It introduces a new approach to establish global solutions for inhomogeneous hyperbolic systems without relying on equilibrium assumptions or source dependence on the state.

Findings

Global existence of entropy weak solutions is proven.

Results apply to systems with inhomogeneous flux and source.

No assumptions on equilibrium solutions or source dependence are needed.

Abstract

General hyperbolic systems of balance laws with inhomogeneous flux and source are studied. Global existence of entropy weak solutions to the Cauchy problem is established for small $BV$ data under appropriate assumptions on the decay of the flux and the source with respect to space and time. There is neither a hypothesis about equilibrium solution nor about the dependence of the source on the state vector as previous results have assumed.