Multi-dimensional scalar balance laws with discontinuous flux
  We consider the problem of existence of entropy weak solutions to scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect both to the space variable x and the unknown quantity u. The problem is formulated in the framework of multi-valued mappings. We use the notion of entropy-measure valued solutions to prove the so-called contraction principle and comparison principle. 
Piotr Gwiazda, Agnieszka Swierczewska-Gwiazda, Petra Wittbold,   Aleksandra Zimmermann
math.AP