  We address the global-in-time existence, stability and long time behaviour of
weak solutions of the three-dimensional compressible Navier-Stokes equations
with potential force. We show the details of the $\alpha$-dependence of
different smoothing rates for weak solutions near $t=0$ under the assumption on
the initial velocity $u_0$ that $u_0\in H^\alpha$ for
$\alpha\in(\frac{1}{2},1]$ and obtain long time convergence of weak solutions
in various norms. We then make use of the Lagrangean framework in comparing the
instantaneous states of corresponding fluid particles in two different
solutions. The present work provides qualitative results on the long time
behaviour of weak solutions and how the weak solutions depend continuously on
initial data and steady states.
