Strong relaxation of the isothermal Euler system to the heat equation

We consider the system of isothermal Euler Equations with a strong damping. For large BV solutions, we show that the density converges to the solution to the heat equation when the friction coefficient $ε^{−1}$ tends to infinity. Our estimates are already valid for small time, including in the initial layer. They are global in space (and even in time when the limits of the density are the same at $±∞$) and they provide rates of convergence when $ε→0$.