I studied that a measure valued solution of a conservation law $u_t+f(u)_x=0$ is a measurable map $\eta : y \rightarrow \eta_y \in Prob(\mathbb{R^n})$ which satisfies $\partial_t(\eta_y, \lambda) +\partial_x (\eta_y, f(\lambda)) =0$ in the sense of distribution on $\mathbb{R^2_+}$. In particular, when the conservation law admits $L^{\infty}$ solution then $\eta_y=\delta_{u(y)}$.
Now I am trying to read "Delta-shock Wave Type Solution of Hyperbolic Systems of Conservation Laws" by Danilov and Shelkovich [1]. In this article
- what do they mean by $\delta$-shocks?
- in which sense these $\delta$-shocks are different from the shocks of the conservation laws?
According to the definition which I stated in the begining any shock solution $u \in L^{\infty}$ can be written as a dirac measure $\eta_y=\delta_{u(y)}$. So:
- are all shocks $\delta$-shocks?
Please suggest me the reference
[1] V.G. Danilov, V.M. Shelkovich (2005): "Delta-shock wave type solution of hyperbolic systems of conservation laws", Quart. Appl. Math. 63, 401-427. doi:10.1090/S0033-569X-05-00961-8