Approximation of the relaxed Euler system at the level of the compressible Navier-Stokes equations

We study the global well-posedness of the Cauchy problem in $\mathbb R^d$, $d\geq2$, for the relaxed compressible Euler system with a Maxwell-type constitutive law for the stress tensor. We show that the corresponding compressible Navier-Stokes solutions serve as both relaxation limits and large-time asymptotic profiles of the global relaxed Euler solutions constructed here. Given a Navier-Stokes background satisfying suitable boundedness and decay assumptions, we construct global smooth relaxed Euler solutions for each fixed $α\in(0,1]$ and sufficiently small $\varepsilon>0$. Here, the background need not be small. After subtracting the stress initial layer, the approximation error is $O(\varepsilon^α)$ uniformly in time. Under suitable additional conditions on the initial perturbations, this error decays in $L^2$ faster than the decay of standard Navier-Stokes background, uniformly in $\varepsilon$, identifying the background solution as a large-time asymptotic profile. As an application, we obtain the corresponding small-data results using classical Navier-Stokes theory to verify the background assumptions.