Global classical solution to 1D compressible Navier–Stokes equations with no vacuum at infinity

Abstract

In this paper, we are concerned with the 1D Cauchy problem of the compressible Navier-Stokes equations with the viscosity . / D 1 C ˇ.ˇ 0/. The initial density can be arbitrarily large and keep a non-vacuum state N > 0 at far fields. We will establish the global existence of the classical solution for 0 ġ< via a priori estimates when the initial density contains vacuum in interior interval or is away from the vacuum. We will show that the solution will not develop vacuum in any finite time if the initial density is away from the vacuum. To study the well-posedness of the problem, it is crucial to obtain the upper bound of the density. Some new weighted estimates are applied to obtain our main results.

On non-resistive limit of 1D MHD equations with no vacuum at infinity

“…Moreover, if ignoring the magnetic field, then MHD system reduces to the compressible Navier-Stokes equations. So, Theorems 1.1 and 1.2 can be seen as an extension of that in [24].…”

“…The upper bound of the density ρ can be shown in the similar manner as that in [24]. However, for completeness of the paper, we give the details here.…”

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

“…However, both the uniform upper bound of density and the large time behavior of solutions are not obtained in [23].…”

“…Recently, for general µ, we [17] establish not only the global existence but also the large-time behavior for classical solutions containing vacuum to the initial boundary value problem for 1D compressible Navier-Stokes equations. For the Cauchy problem (1.1)-(1.3) without external force (f = 0), Ye [23] studies the global classical large solutions under the following restriction on µ(ρ): µ(ρ) = 1 + ρ β , 0 ≤ β < γ.…”

Analytical Solution to 1D Compressible Navier-Stokes Equations

“…Ding et al [18] got the global existence of classical solutions to 1D compressible Navier-Stokes equations in bounded domains, provided that μ ∈ C 2 ½0,+∞Þ satisfies 0 < μ ≤ μðρÞ ≤ Cð1 + PðρÞÞ. 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.…”