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

TL;DR

This paper analyzes the vanishing viscosity limits of a 1D zero-pressure gas dynamics system with initial data consisting of delta distributions at two points, using asymptotic analysis and Hopf-Cole transformations.

Contribution

It provides a detailed structure of the vanishing viscosity limits for initial delta-distributions in a 1D gas dynamics system, extending understanding of such limits with multiple point sources.

Findings

Explicit characterization of the vanishing viscosity limit structure.

Use of asymptotic properties of the erfc function in analysis.

Application of Hopf-Cole transformations to analyze solutions.

Abstract

In this article, we consider the one-dimensional zero-pressure gas dynamics system \[ u_t + \left( {u^2}/{2} \right)_x = 0,\ \rho_t + (\rho u)_x = 0 \] in the upper-half plane with a linear combination of two $\delta$-distributions \[ u|_{t=0} = u_a\ \delta_{x=a} + u_b\ \delta_{x=b},\ \rho|_{t=0} = \rho_c\ \delta_{x=c} + \rho_d\ \delta_{x=d} \] 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 \[ u^\epsilon_t + \left({(u^\epsilon)^2}/{2} \right)_x =\frac{\epsilon}{2} u^\epsilon_{xx},\ \rho^\epsilon_t + (\rho^\epsilon u^\epsilon)_x = \frac{\epsilon}{2} \rho^\epsilon_{xx}. \] For this purpose, we extensively use the various asymptotic properties of the function erfc$: z \longmapsto \int_{z}^{\infty} e^{-s^2}\ ds$ along with suitable Hopf-Cole transformations.