  We consider the isothermal Euler system with damping. We rigorously show the
convergence of Barenblatt solutions towards a limit Gaussian profile in the
isothermal limit $\gamma$ $\rightarrow$ 1, and we explicitly compute the
propagation and the behavior of Gaussian initial data. We then show the weak L
1 convergence of the density as well as the asymptotic behavior of its first
and second moments. Contents 1. Introduction 1 2. Assumptions and main results
3 3. The limit $\gamma$ $\rightarrow$ 1 of Barenblatt's solutions 6 4. Gaussian
solutions 9 5. Evolution of certain quantities 10 6. Convergence 15 7.
Conclusion 17 References 17
