Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains

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.

Analytical Solution to 1D Compressible Navier-Stokes Equations

“…Ye [19] obtained the global classical large solutions to the Cauchy problem (1) and (2) with the restriction μðρÞ = 1 + ρ β ; 0 ≤ β < γ. Zhang and Zhu [20] derived the global existence of classical solution to the initial boundary value problem for the onedimensional Navier-Stokes equations for viscous compressible and heat-conducting fluids in a bounded domain with the Robin boundary condition on temperature. Li et al [21] derive the uniform upper bound of density and the global wellposedness of strong (classical) large solutions to the Cauchy problem with the external force. For two-dimensional case, global well-posedness of classical solutions to the Cauchy problem or periodic domain problem of compressible Navier-Stokes equations with vacuum was obtained in [22][23][24] when the first and second viscosity coefficients are μ and λðρÞ, respectively.…”