  For periodic initial data with initial density allowed to vanish, we
establish the global existence of strong and weak solutions for the
two-dimensional compressible Navier-Stokes equations with no restrictions on
the size of initial data provided the bulk viscosity coefficient is $\lambda =
\rho^{\beta}$ with $\beta>4/3$. These results generalize and improve the
previous ones due to Vaigant-Kazhikhov([Sib. Math. J. (1995), 36(6),
1283-1316]) which requires $\beta>3$. Moreover, both the uniform upper bound of
the density and the large-time behavior of the strong and weak solutions are
also obtained.
