Explicit structure of the vanishing viscosity limits for the zero-pressure gas dynamics system initiated by the linear combination of a characteristic function and a $\delta$-distribution

In this article, we consider the one-dimensional zero-pressure gas dynamics system $$u_t+{\left(u^2\mathrm{/}2\right)}_x=0,{\rho }_t+(\rho u{)}_x=0$$ in the upper-half plane with a linear combination of a characteristic function and a $\delta$-measure $$u{\mathrm{\mid }}_{t=0}=u_a{\chi }_{{}_{(-\mathrm{\infty },a)}}+u_b{\delta }_{x=b},\rho {\mathrm{\mid }}_{t=0}={\rho }_c{\chi }_{{}_{(-\mathrm{\infty },c)}}+{\rho }_d{\delta }_{x=d}$$ as initial data, where $a$, $b$, $c$, $d$ are distinct points on the real line ordered as $a<c<b<d$, and provide a detailed analysis of the vanishing viscosity limits for the above system utilizing the corresponding modified adhesion model $$u_t^{\epsilon }+{\left((u^{\epsilon }{)}^2\mathrm{/}2\right)}_x=\frac{\epsilon }{2}{u}_{xx}^{\epsilon },{\rho }_t^{\epsilon }+({\rho }^{\epsilon }{u}^{\epsilon }{)}_x=\frac{\epsilon }{2}{\rho }_{xx}^{\epsilon }\mathrm{.}$$ For this purpose, we use suitable Hopf-Cole transformations and various asymptotic properties of the function erfc$:z⟼{\int }_z^{\mathrm{\infty }}{e}^{-s^2}ds$.