A nonlinear conservation law with a step function involving the flux: A distributional approach

The present paper concerns a Riemann problem for a conservation law with a nonlinear flux involving a step function. In a convenient space of distributions, we will see the emergence of a delta standing wave which extends the corresponding result, obtained by Sun in 2013 and also by Shen in 2014, for a linear flux. In the nonlinear setting, a new phenomenon arises: starting from a vanishing initial condition, the concentration of matter in a fixed point becomes possible. These results are easy to get essentially because, in our approach, the product of distributions is a distribution that does not depend of approximation processes.