Global well-posedness and Newtonian limit of the compressible upper-convected Maxwell model in analytic spaces

In this paper, we investigate the compressible upper-convected Maxwell (UCM) system in ${\mathbb{R}}^3$, which corresponds to the purely elastic case of the compressible Oldroyd-B model with vanishing retardation time. In this regime, the velocity equation loses the viscous dissipation term, and the constitutive nonlinearities induce a derivative loss, so that the standard approaches in Sobolev or finitely regular Besov spaces fail. We therefore work in analytic function spaces, realized as exponentially weighted hybrid Besov spaces, and develop a global-in-time analytic framework based on Littlewood–Paley analysis and uniform-in-$\tau$ energy estimates. A key ingredient is the introduction of the kinematic stress tensor, which allows us to establish the required a priori estimates in hybrid Besov spaces. Based on these estimates, we prove the existence of a unique global-in-time analytic solution for small analytic initial data. Moreover, for ill-prepared data, we rigorously justify the global-in-time relaxation (Newtonian) limit as the relaxation parameter $\tau \to 0$, with an explicit convergence rate of order $O(\sqrt{\tau })$.