Asymptotic approximation of a modified compressible Navier-Stokes system

TL;DR

This paper analyzes the long-term behavior of a modified compressible Navier-Stokes system, introducing new decomposition and approximation methods that enable explicit asymptotic descriptions of solutions.

Contribution

It presents a novel decomposition of the momentum field and a Hermite function-based approximation technique for the heat equation, advancing understanding of the system's asymptotics.

Findings

Established existence of solutions to the mcNS system.

Hermite function approximation captures leading order asymptotics.

Explicit evaluation of asymptotic terms under certain conditions.

Abstract

We study the long time asymptotics of a modified compressible Navier-Stokes system (mcNS) inspired by the previous work of Hoff and Zumbrun. We introduce a new decomposition of the momentum field into its irrotational and incompressible parts, and a new method for approximating solutions of the heat equation 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 and show that the approximation in terms of Hermite functions gives the leading order terms in the long-time asymptotics, and under certain assumptions can be evaluated explicitly.