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# Initial value problem for the nonconservative zero-pressure gas dynamics system

## Problème de la valeur initiale pour le système dynamique des gaz à pression nulle non conservateur

Abhishek Das
Tata Institute of Fundamental Research
India

K. T. Joseph
Tata Institute of Fundamental Research
India

Manas R. Sahoo
National Institute of Science Education and Research
India

Published on 5 January 2021   DOI : 10.21494/ISTE.OP.2020.0580

### Mots-clés

In this article, we study initial value problem for the zero-pressure gas dynamics system in non conservative form and the associated adhesion approximation. We use adhesion approximation and modi-ed adhesion approximation in the construction of weak asymptotic solution. First we prove a general existence result for the adhesion model for the initial velocity component in $H^s \mbox{ for } s$ > $\frac{n}{2} + 1$ and the initial data for the density component being a $C^1$ function. Using this, we construct weak asymptotic solution for the system with initial velocity in $L^2 \cap L^{\infty}$ and the initial density being a bounded Borel measure. Then we make a detailed analysis of the explicit formula for the weak asymptotic solution and generalized solution for the plane-wave type initial data.

In this article, we study initial value problem for the zero-pressure gas dynamics system in non conservative form and the associated adhesion approximation. We use adhesion approximation and modi-ed adhesion approximation in the construction of weak asymptotic solution. First we prove a general existence result for the adhesion model for the initial velocity component in $H^s \mbox{ for } s$ > $\frac{n}{2} + 1$ and the initial data for the density component being a $C^1$ function. Using this, we construct weak asymptotic solution for the system with initial velocity in $L^2 \cap L^{\infty}$ and the initial density being a bounded Borel measure. Then we make a detailed analysis of the explicit formula for the weak asymptotic solution and generalized solution for the plane-wave type initial data.