Gevrey Analyticity and Decay for the Compressible Navier-Stokes System with Capillarity

We are concerned with a system of equations governing the evolution of isothermal, viscous, and capillary compressible fluids, which can be used as a phase transition model. We prove that the global solutions with critical regularity that have been constructed in [11] by the second author and B. Desjardins are Gevrey analytic, then 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, and Tadmor [2] dedicated to the classical incompressible Navier-Stokes equations, and requires us to establish new bilinear estimates (of independent interest) involving the Gevrey regularity for the product or composition of functions. Our approach is partly inspired by the work of Bae, Biswas, and Tadmor [2] dedicated to the classical incompressible Navier-Stokes equations, and requires us to establish new bilinear estimates (of independent interest) involving the Gevrey regularity for the product or composition of functions. To the best of our knowledge, our work is the first one that exhibits Gevrey analyticity for a model of compressible fluids.