Global strong solutions to 1-D vacuum free boundary problem for compressible Navier–Stokes equations with variable viscosity and thermal conductivity

Abstract

In this paper, we investigate the vacuum free boundary problem of one-dimensional heat-conducting compressible Navier–Stokes equations where the viscosity coefficient depends on the density, and the heat conductivity coefficient depends on the temperature, satisfying a physical assumption from the Chapman–Enskog expansion of the Boltzmann equation. The fluid connects to the vacuum continuously, thus the system is degenerate near the free boundary. The global existence and uniqueness of strong solutions for the free boundary problem are established when the initial data are large. The result is proved by using both the Lagrangian mass coordinate and the Lagrangian trajectory coordinate. An key observation is that the Jacobian between these coordinates are bounded from above and below by positive constants.