  We investigate the Navier-Stokes-Cattaneo-Christov (NSC) system in
$\mathbb{R}^d$ ($d\geq3$), a model of heat-conductive compressible flows
serving as a finite speed of propagation approximation of the
Navier-Stokes-Fourier (NSF) system. Due to the presence of Oldroyd's
upper-convected derivatives, the system (NSC) exhibits a \textit{lack of
hyperbolicity} which makes it challenging to establish its well-posedness,
especially in multi-dimensional contexts. In this paper, within a critical
regularity functional framework, we prove the global-in-time well-posedness of
(NSC) for initial data that are small perturbations of constant equilibria,
uniformly with respect to the approximation parameter $\varepsilon>0$. Then,
building upon this result, we obtain the sharp large-time asymptotic behaviour
of (NSC) and, for all time $t>0$, we derive quantitative error estimates
between the solutions of (NSC) and (NSF). To the best of our knowledge, our
work provides the first strong convergence result for this relaxation procedure
in the three-dimensional setting and for ill-prepared data.
  The (NSC) system is partially dissipative and incorporates both partial
diffusion and partial damping mechanisms. To address these aspects and ensure
the large-time stability of the solutions, we construct localized-in-frequency
perturbed energy functionals based on the hypocoercivity theory. More
precisely, our analysis relies on partitioning the frequency space into
\textit{three} distinct regimes: low, medium and high frequencies. Within each
frequency regime, we introduce effective unknowns and Lyapunov functionals,
revealing the spectrally expected dissipative structures.
