New entropy conditions for the scalar conservation law with discontinuous flux
Abstract: We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux in rather general form (no convexity or genuine linearity is needed). We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under the assumption that initial data belong to the BV-class. Such initial data enable us to prove that the sequence of solutions to a special vanishing viscosity approximation of the considered equation is, at the same time, the sequence of quasisolutions to a non-degenerate scalar conservation law. This provides existence of the solution admitting strong traces at the interface. The admissibility conditions are chosen so that a kind of crossing condition is satisfied which, together with existence of traces, provides uniqueness of the solution.