The vanishing viscosity limit for Hamilton-Jacobi equations on Networks
  For a Hamilton-Jacobi equation defined on a network, we introduce its vanishing viscosity approximation. The elliptic equation is given on the edges and coupled with Kirchhoff-type conditions at the transition vertices. We prove that there exists exactly one solution of this elliptic approximation and mainly that, as the viscosity vanishes, it converges to the unique solution of the original problem. 
Fabio Camilli, Claudio Marchi, Dirk Schieborn
math.AP