Global existence of classical solutions for two-dimensional isentropic compressible Navier–Stokes equations with small initial mass

In this paper, we consider the initial-boundary value problem of two-dimensional isentropic compressible Navier–Stokes equations with vacuum on the square domain. Based on the time-weighted uniform estimates, we prove that the classical solution exists globally in time if the initial mass $\|\rho_{0}\|_{L^{1}}$ of the fluid is small. Here, we do not require the initial energy or the upper bound of the initial density to be small.