Global large smooth solutions for isothermal Euler equations with damping and small parameter

This paper concerns smooth solutions to Cauchy problem for isothermal Euler equations with damping depending on a relaxation time. We prove that the problem admits a unique solution when either the relaxation time or the initial datum is sufficiently small. In particular, this yields the global existence of a large smooth solution when the relaxation time is sufficiently small. We justify that, in an appropriate time scaling, the density of Euler equations with damping converges to the large solution of the heat equation as the relaxation time tends to zero. Moreover, we establish error estimates of such a convergence for the large solutions. A key step in proving these results is a uniform estimate of a quantity related to Darcy's law.