On unified theory for scalar conservation laws with fluxes and sources discontinuous with respect to the unknown
Abstract
We deal with the Cauchy problem for multi-dimensional scalar conservation laws, where the fluxes and the source terms can be discontinuous functions of the unknown. The main novelty of the paper is the introduction of a kinetic formulation for the considered problem.
To handle the discontinuities we work in the framework of re-parametrization of the flux and the source functions, which was previously used for Kruikov entropy solutions. Within this approach we obtain a fairly complete picture: existence of entropy measure valued solutions, entropy weak solutions and their equivalence to the kinetic solution.
The results of existence and uniqueness follow under the assumption of Holder continuity at zero of the flux. The source term, what is another novelty for the studies on problems with discontinuous flux, is only assumed to be one-side Lipschitz, not necessarily monotone function.