We investigate the Cauchy problem for the Navier–Stokes equations for viscous compressible fluids with capillarity. The linear third-order capillarity term behaves like the heat diffusion of density fluctuations, which enables us to provide an equivalent characterization of Gevrey analyticity, optimal decay, and initial regularity criteria in \(\mathbb {R}^d\) ( \(d \ge 2\) ). Precisely, in the critical \(L^p\) framework, it is proved that the Besov space \(\dot{B}^{\sigma _1}_{2,\infty }\) -boundedness condition (with \(d/2-2d/p\le \sigma _1<d/2-1\) ) of the low-frequency part of initial perturbation is not only sufficient, but also necessary to achieve Gevrey analyticity or upper bounds of decay estimates for arbitrary higher-order derivatives. Furthermore, we show that the upper and lower bounds of decay estimates hold if and only if the low-frequency part of initial perturbation belongs to a nontrivial subset of \(\dot{B}^{\sigma _1}_{2,\infty }\) . Our approach is partly inspired by Hoff-Zumbrun’s spectral analysis, and requires us to establish the uniform bounds on the growth of the radius of analyticity and the faster decay of the discrepancy between the nonlinear solution and the diffusion profile.
