Gevrey analyticity and decay for the compressible Navier-Stokes system with capillarity

We are concerned with an isothermal model of viscous and capillary compressible fluids derived by J. E. Dunn and J. Serrin (1985), which can be used as a phase transition model. Compared with the classical compressible Navier-Stokes equations, there is a smoothing effect on the density that comes from the capillary terms. First, we prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins (2001), are Gevrey analytic. Second, we extend that result to a more general critical L p framework. As a consequence, we obtain algebraic time-decay estimates in critical Besov spaces (and even exponential decay for the high frequencies) for any derivatives of the solution. Our approach is partly inspired by the work of Bae, Biswas \& Tadmor [2] dedicated to the classical incompressible Navier-Stokes equations, and requires our establishing new bilinear estimates (of independent interest) involving the Gevrey regularity for the product or composition of functions. To the best of our knowledge, this is the first work pointing out Gevrey analyticity for a model of compressible fluids.