Global Solutions to the 2D Compressible Navier-Stokes Equations with Some Large Initial Data

We consider the global well-posedness of strong solutions to the Cauchy problem of compressible barotropic Navier-Stokes equations in Double-struck capital R-2. By exploiting the global-in-time estimate to the two-dimensional (2D for short) classical incompressible Navier-Stokes equations and using techniques developed in (SIAM J Math Anal, 2020, 52(2): 1806-1843), we derive the global existence of solutions provided that the initial data satisfies some smallness condition. In particular, the initial velocity with some arbitrary Besov norm of potential part DOUBLE-STRUCK CAPITAL Pu-0 and large high oscillation are allowed in our results. Moreover, we also construct an example with the initial data involving such a smallness condition, but with a norm that is arbitrarily large.