Existence and Blowup Behavior of Global Strong Solutions to the Two-Dimensional Baratropic Compressible Navier-Stokes System with Vacuum and Large Initial Data

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.Comment: 31 page