Global classical solutions of compressible isentropic Navier-Stokes equations with small density

This paper concerns the Cauchy problem of compressible isentropic Navier-Stokes equations in the whole space R3. First, we show that if ρ0∈Lγ∩H3, then the problem has a unique global classical solution on R3×[0,T] with any T∈(0,∞), provided the upper bound of the initial density is suitably small and the adiabatic exponent γ∈(1,6). If, in addition, the conservation law of the total mass is satisfied (i.e., ρ0∈L1), then the global existence theorem with small density holds for any γ>1. It is worth mentioning that the initial total energy can be arbitrarily large and the initial vacuum is allowed. Thus, the results obtained particularly extend the one due to Huang-Li-Xin (Huang et al., 2012), where the global well-posedness of classical solutions with small energy was proved.