English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this article, we consider the one-dimensional zero-pressure gas dynamicssystem \[ u_t + \left( {u^2}/{2} \right)_x = 0,\ \rho_t + (\rho u)_x = 0 \] inthe upper-half plane with a linear combination of a characteristic function anda $\delta$-measure \[ u|_{t=0} = u_a\ \chi_{ {}_{ \left( -\infty , a \right) }} + u_b\ \delta_{x=b},\ \rho|_{t=0} = \rho_c\ \chi_{ {}_{ \left( -\infty , c\right) } } + \rho_d\ \delta_{x=d} \] as initial data, where $a$, $b$, $c$, $d$are distinct points on the real line ordered as $a < c < b < d$, and provide adetailed analysis of the vanishing viscosity limits for the above systemutilizing 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 use suitable Hopf-Coletransformations and various asymptotic properties of the function erfc$: z\longmapsto \int_{z}^{\infty} e^{-s^2}\ ds$.

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