Diagonal hyperbolic systems with large and monotone data Part I: Global continuous solutions

TL;DR

This paper proves the global existence of continuous solutions for large, monotone initial data in diagonal hyperbolic systems, including non-strictly hyperbolic cases, relevant to dislocation density modeling.

Contribution

It introduces a new gradient entropy estimate to establish global solutions for large, non-decreasing initial data in hyperbolic systems, even when not strictly hyperbolic.

Findings

Global existence of continuous solutions proven

Applicable to non-strictly hyperbolic systems

Relevant to dislocation density dynamics

Abstract

In this paper, we study diagonal hyperbolic systems in one space dimension. Based on a new gradient entropy estimate, we prove the global existence of a continuous solution, for large and non-decreasing initial data. We remark that these results cover the case of systems which are hyperbolic but not strictly hyperbolic. Physically, this kind of diagonal hyperbolic systems appears naturally in the modelling of the dynamics of dislocation densities.