Delta Shock Waves in the Shallow Water System

Abstract

We consider a Riemann problem for the shallow water system $u_{t} +\big (v+\textstyle \frac{1}{2}u^{2}\big )_{x}=0$, $v_{t}+\big (u+uv\big )_{x}=0$ and evaluate all singular solutions of the form $u(x,t)=l(t)+b(t)H\big (x-\gamma (t)\big )+a(t)\delta \big (x-\gamma (t)\big )$, $v(x,t)=k(t)+c(t)H\big (x-\gamma (t)\big )$, where $l,b,a,k,c,\gamma :\mathbb {R}\rightarrow \mathbb {R}$ are $C^{1}$-functions of time t, H is the Heaviside function, and $\delta$ stands for the Dirac measure with support at the origin. A product of distributions, not constructed by approximation processes, is used to define a solution concept, that is a consistent extension of the classical solution concept. Results showing the advantage of this framework are briefly presented in the introduction.