We construct global smooth approximate solution to a scalar conservation law with arbitrary smooth monotonic initial data. Different kinds of singularities interactions which arise during the evolution of the initial data are described as well. In order to solve the problem, we use and further develop the weak asymptotic method, recently introduced technique for investigating nonlinear waves interactions.
"Smooth Approximations of Global in Time Solutions to Scalar Conservation Laws." Abstr. Appl. Anal. 2009 1 - 26, 2009. https://doi.org/10.1155/2009/350762