Asymptotic Approximation of a Modified Compressible Navier-Stokes System

We study the effects of localization on the long-time asymptotics of a modified compressible Navier-Stokes system (mcNS) inspired by the previous work of Hoff and Zumbrun [4]. We introduce a new decomposition of the momentum field into its irrotational and incompressible parts, and a new method for approximating solutions of jointly hyperbolic-parabolic equations in terms of Hermite functions in which n(th) order approximations can be computed for solutions with n(th)-order moments. We then obtain existence of solutions to the mcNS system in weighted spaces and, based on the decay rates obtained for the various pieces of the solutions, determine the optimal choice of asymptotic approximation with respect to the various localization assumptions, which in certain cases can be evaluated explicitly in terms of Hermite functions.