On the Cauchy problem for a one-dimensional compressible viscous polytropic ideal gas

Abstract

In this paper, we first prove the regularity and continuous dependence on initial data for $H^i$-solutions $(i=1,2,4)$ for large initial data and then show the large-time behavior of $H^i(i=2,4)$-global solutions for small initial data to the Cauchy problem for the compressible Navier--Stokes equations of a one-dimensional viscous polytropic ideal gas. Moreover, we also obtain the large-time behavior of ``small'' classical solutions {\em in the norm of classical solutions} for this model.