Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

TL;DR

This paper compares how solutions to two different compressible approximations of the Navier-Stokes equations decay over time, revealing the impact of nonlinear damping versus advection-like terms on decay rates.

Contribution

It introduces a method using decay character of initial data to compare energy decay rates of different compressible Navier-Stokes approximations, highlighting the influence of nonlinear damping.

Findings

Systems with nonlinear damping decay slower than those with advection-like terms.

Decay behavior depends on initial data, with some driven by the difference from linear solutions.

Characterization of initial data sets influencing decay mechanisms.

Abstract

In this article, we use the decay character of initial data to compare the energy decay rates of solutions to different compressible approximations to the Navier-Stokes equations. We show that the system having a nonlinear damping term has slower decay than its counterpart with an advection-like term. Moreover, we characterize a set of initial data for which the decay of the first system is driven by the difference between the full solution and the solution to the linear part, while for the second system the linear part provides the decay rate.