Global continuous solutions to diagonalizable hyperbolic systems with large and monotone data

TL;DR

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

Contribution

Introduces a new gradient entropy estimate to establish global solutions for large, monotone data in diagonalizable hyperbolic systems, extending to non-strictly hyperbolic cases.

Findings

Global existence of continuous solutions proven

Results cover non-strictly hyperbolic systems

Applicable to dislocation density dynamics

Abstract

In this paper, we study diagonalizable 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 nondecreasing initial data. Moreover, we show in particular cases some uniqueness results. We also remark that these results cover the case of systems which are hyperbolic but not strictly hyperbolic. Physically, this kind of diagonalizable hyperbolic systems appears naturally in the modelling of the dynamics of dislocation densities.