Dynamical behaviors for 1D compressible Navier–Stokes equations with density-dependent viscosity

Abstract

The dynamical behaviors of vacuum states for one-dimensional compressible Navier–Stokes equations with density-dependent viscosity coefficient are considered. It is first shown that a unique strong solution to the free boundary value problem exists globally in time, the free boundary expands outwards at an algebraic rate in time, and the density is strictly positive in any finite time but decays pointwise to zero time-asymptotically. Then, it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum. The solution is piecewise regular and contains a piece of continuous vacuum before the time T ∗ > 0 , which is compressed at an algebraic rate and vanishes at the time T ∗ , meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially.