Global existence for a class of large solution to compressible Navier–Stokes equations with vacuum

In this paper, we are concerned with the Cauchy problem of the three-dimensional isentropic compressible Navier–Stokes equations. We prove the global existence and uniqueness of classical solutions with large initial energy and vacuum, under the assumptions that the fluid is nearly isothermal (i.e., the adiabatic exponent γ is sufficiently close to 1), and that the far-field density ρ ~ is either vacuum or close to vacuum. To the best of our knowledge, we establish the first result on the global existence of large-energy solutions with vacuum to the three-dimensional compressible Navier–Stokes equations for the cases of vacuum and nonvacuum far-field constant states, which generalizes the result by Huang, Li and Xin (Commun Pure Appl Math 65:549–585, 2012) on classical solutions with vacuum and small energy (large oscillations).