Entropy solutions to a non-conservative and non-strictly hyperbolic diagonal system inspired by dislocation dynamics

In this work, we study the existence of solutions to a 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\times 2$$\end{document} non-conservative and non-strictly hyperbolic system in one space dimension related to the dynamics of dislocation densities in crystallography, propagating in two opposite directions. For such systems, existence results are mainly established in the sense of viscosity solutions for Hamilton-Jacobi equations. We study this problem for large initial data using the notions of the theory of conservation laws by constructing entropy solutions through the means of an adapted Godunov scheme, where the associated Riemann problem enjoys new features, more elementary waves than usual, and loss of uniqueness in many cases. The existence is obtained in spaces of functions with bounded fractional total variation BVs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$BV^s$$\end{document}, for all 0<s≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<s\le 1$$\end{document}, such that BV1=BV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$BV^1=BV$$\end{document}.