A Note on Large Time Behavior of Velocity in the Baratropic Compressible Navier-Stokes Equations

TL;DR

This paper discusses the large-time behavior of velocity in the compressible Navier-Stokes equations, showing convergence to an equilibrium velocity in H^1 norm, correcting previous results and extending understanding of long-term dynamics.

Contribution

It proves that velocity converges to a unique equilibrium velocity in H^1 norm, refining previous large-time behavior results for solutions with vanishing initial density.

Findings

Velocity converges to equilibrium velocity in H^1 norm.

The equilibrium velocity is uniquely determined by initial data.

Provides correction to earlier large-time behavior results.

Abstract

Recently, for periodic initial data with initial density allowed to vanish, Huang and Li [1] establish the global existence of strong and weak solutions for the two-dimensional compressible Navier{Stokes equations with no restrictions on the size of initial data provided the bulk viscosity coefficient is \lambda = \rho^\beta with \beta > 4/3. Moreover, the large-time behavior of the strong and weak solutions are also obtained, in which the velocity gradient strongly converges to zero in L^2 norm. In this note, we further point out that the velocity strongly converges to an equilibrium velocity in H^1 norm, in which the equilibrium velocity is uniquely determined by the initial data. Our result can also be regarded a correction for the result of large-time behavior of velocity in [2].