Scalar conservation laws with discontinuous flux: existence and uniqueness

Abstract

We study the well-posedness of the Cauchy problem for scalar conservation laws with discontinuous, non-degenerate fluxes. Locally, the fluxes are piecewise smooth across interfaces described by a Heaviside-type discontinuity, with left and right states depending smoothly on both space and the solution variable. The interface is given by a smooth function, and the fluxes vanish at the boundary values of the admissible interval for the solution. In addition, we consider the more general case of heterogeneous flux functions with bounded variation in the spatial variable and smooth dependence on the solution variable, again vanishing at the prescribed boundary states. For this setting, we construct a stable semigroup of solutions, thereby establishing a well-posed solution framework.