GLOBAL SOLUTIONS TO THE ISENTROPIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH A CLASS OF LARGE INITIAL DATA

this paper, we consider the global well-posedness problem of the isentropic compressible Navier-Stokes equations in the whole space R-N, with N >= 2. It is shown that the one-parameter group e(+/- it Lambda) is of great importance to the behaviors of solutions to the isentropic compressible Navier-Stokes equations in the low frequency part. In order to better reflect the dispersive property of this system, we introduce a new solution space that characterizes the behaviors of the solutions in different frequencies and prove that the isentropic compressible Navier-Stokes equations admit global solutions when the initial data are close to a stable equilibrium in the sense of suitable hybrid Besov norms. As a consequence, the initial velocity with an arbitrary (B) over dot(2,1)(N/2-1) norm of potential part P(perpendicular to)u(0) and a large highly oscillating initial velocity are allowed in our results. The proof relies heavily on the dispersive estimates for the system of acoustics and a careful study of the nonlinear terms.