Global existence of classical solutions to two-dimensional Navier-Stokes equations with Cauchy data containing vacuum

In this paper, the Cauchy problem to the two-dimensional isentropic compressible Navier-Stokes equations with smooth initial data containing vacuum is investigated. If the initial data are of small energy but possibly large oscillations, we obtain the global well-posedness of classical solutions in the case of initially nonvacuum far fields. In particular, the smallness of the energy only depends on the H-beta(0<beta <= 1) norm of the initial velocity, where beta can be arbitrary close to 0. In the case of compactly supported initial density, a blow-up example is given. Copyright (c) 2013 John Wiley & Sons, Ltd.