Delta shock waves in conservation laws with impulsive moving source: some results obtained by multiplying distributions

Abstract

The present paper concerns the study of a Riemann problem for the conservation law ut + [ø(u)]x = kδ(x-vt) where x, t, k, v and u = u(x,t) are real numbers. We consider ø an entire function taking real values on the real axis and δ stands for the Dirac measure. Within a convenient space of distributions we will explicitly see the possible emergence of waves with the shape of shock waves, delta waves and delta shock waves. For this purpose, we define a rigorous concept of a solution which extends both the classical solution concept and a weak solution concept. All this framework is developed in the setting of a distributional product that is not constructed by approximation. We include the main ideas of this product for the reader’s convenience. Recall that delta shock waves are relevant physical phenomena which may be interpreted as processes of concentration of mass or even as processes of formation of galaxies in the universe.