LAUSR

Abstract This paper concerns the Cauchy problem of compressible isentropic Navier–Stokes equations in the whole space R 3 . First, we show that if ρ 0 ∈ L γ ∩ H 3 , then the problem has a unique global classical solution on R 3 × [ 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., ρ 0 ∈ L 1 ), 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.