Multiplication of distributions and singular waves in several physical models

Abstract

AbstractWe study of a Riemann problem for a one-dimensional $$2\hspace{1.111pt}{\times }\hspace{1.111pt}2$$ 2 × 2 system of conservation laws, which includes some particular cases of known physical systems, such as the pressureless gas dynamics system, Euler equations for isentropic flow, the Brio system, and the shallow water system. The main results of this study reveal the emergence of shock waves and delta shock waves as explicit solutions. We use the concept of $$\alpha $$ α -solution defined in the setting of a product of distributions not defined by approximation. Notably, this study does not assume any classical results about conservation laws, presenting a simpler and more general framework for constructing singular solutions to other equations or systems. Additionally, this paper includes several comments on the four physical systems mentioned, along with formulas for multiplying distributions and evaluating the compositions of a function with a distribution.