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

TL;DR

This paper proves that solutions to a capillarity-enhanced compressible Navier-Stokes model are Gevrey analytic and exhibit decay properties, extending previous results and establishing new bilinear estimates in a critical regularity framework.

Contribution

It demonstrates Gevrey analyticity of solutions for a compressible fluid model with capillarity and extends decay estimates to a broader critical Lp setting.

Findings

Solutions are Gevrey analytic with critical regularity.

Algebraic and exponential decay estimates are established.

New bilinear estimates involving Gevrey regularity are developed.

Abstract

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.