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In this paper, the Cauchy problem for the three-dimensional (3-D) isentropiccompressible Navier-Stokes equations with degenerate viscosities is considered.By introducing some new variables and making use of the "quasi-symmetrichyperbolic"-"degenerate elliptic" coupled structure to control the behavior ofthe fluid velocity, we prove the global-in-time well-posedness of regularsolutions with vacuum for a class of smooth initial data that are of smalldensity but possibly large velocities. Here the initial mass density isrequired to decay to zero in the far field, and the spectrum of the Jacobimatrix of the initial velocity are all positive. The result here applies to aclass of degenerate density-dependent viscosity coefficients, is independent ofthe BD-entropy, and seems to be the first on the global existence of smoothsolutions which have large velocities and contain vacuum state for suchdegenerate system in three space dimensions.

Global well-posedness of regular solutions to the three-dimensional  isentropic compressible Navier-Stokes Equations with degenerate viscosities  and vacuum