Continuous solution for a non-linear eikonal system

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. We prove a global existence result in the framework of continuous viscosity solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity, relying on a new gradient entropy and \begin{document}$ BV $\end{document} estimates. A uniqueness result is also proved through a comparison principle property.