Explicit structure of the vanishing viscosity limits with initial data consisting of $\delta$-distributions starting from two point sources

Abstract

In this article, we consider the one-dimensional zero-pressure gas dynamics system \[ u_t + \left( {u^2}/{2} \right)_x = 0,\ ρ_t + (ρu)_x = 0 \] in the upper-half plane with a linear combination of two $\delta$-distributions \[ u|_{t=0} = u_a\ δ_{x=a} + u_b\ δ_{x=b},\ ρ|_{t=0} = ρ_c\ δ_{x=c} + ρ_d\ δ_{x=d} \] as initial data. Here $a$, $b$, $c$, $d$ are distinct points on the real line ordered as $a<c<b<d$. Our objective is to provide a detailed analysis of the structure of the vanishing viscosity limits of this system utilizing the corresponding modified adhesion model \[ u^ε_t + \left({(u^ε)^2}/{2} \right)_x =\fracε{2} u^ε_{xx},\ ρ^ε_t + (ρ^εu^ε)_x = \fracε{2} ρ^ε_{xx}. \] For this purpose, we extensively use the various asymptotic properties of the function erfc$:z⟼{\int }_z^{\infty }{e}^{-s^2}ds$ along with suitable Hopf-Cole transformations.

52 pages, 14 figures