$ BV $ solution for a non-linear Hamilton-Jacobi system

Abstract

<p style='text-indent:20px;'>In this work, we are dealing with a non-linear eikonal system in one dimensional space that describes the evolution of interfaces moving with non-signed strongly coupled velocities. For such kind of systems, previous results on the existence and uniqueness are available for quasi-monotone systems and other special systems in Lipschitz continuous space. It is worth mentioning that our system includes, in particular, the case of non-decreasing solution where some existence and uniqueness results arose for strictly hyperbolic systems with a small total variation. In the present paper, we consider initial data with unnecessarily small <inline-formula><tex-math id="M2">\begin{document}$ BV $\end{document}</tex-math></inline-formula> seminorm, and we use some <inline-formula><tex-math id="M3">\begin{document}$ BV $\end{document}</tex-math></inline-formula> bounds to prove a global-in-time existence result of this system in the framework of discontinuous viscosity solution.