Global Well-Posedness and Large-Time Behavior of 1D Compressible Navier-Stokes System with Density-Depending Viscosity and Vacuum in Unbounded Domains

TL;DR

This paper establishes the global existence, uniqueness, and boundedness of solutions for 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum, also analyzing their large-time behavior.

Contribution

It proves the global well-posedness and boundedness of solutions with large initial data and vacuum, under general density-dependent viscosity assumptions.

Findings

Existence of unique global strong solutions for large initial data with vacuum.

Density remains bounded uniformly in time.

Characterization of large-time behavior of solutions without external forces.

Abstract

We consider the Cauchy problem for one-dimensional (1D) barotropic compressible Navier-Stokes equations with density-dependent viscosity and large external force. Under a general assumption on the density-dependent viscosity, we prove that the Cauchy problem admits a unique global strong (classical) solution for the large initial data with vacuum. Moreover, the density is proved to be bounded from above time-independently. As a consequence, we obtain the large time behavior of the solution without external forces.