Global solution to 3D spherically symmetric compressible Navier–Stokes equations with large data

Abstract In this paper, we proved the global well-posedness of the solution to 3D spherically symmetric, compressible and isentropic Navier–Stokes equations in the whole space with arbitrarily large initial data when the shear viscosity μ is a positive constant and the bulk viscosity λ ( ρ ) = ρ β with 0 ≤ β ≤ γ and γ > 1 being the adiabatic exponent in the γ -law pressure. First, the global classical solution is obtained away from the symmetry center r = 0 with arbitrarily large and non-vacuum data. In particular, it is shown that the solution will not develop the vacuum states in any finite time away from the symmetry center if the initial density does not contain vacuum states. Then the global weak solutions with the symmetry center r = 0 are obtained as the limit of the classical solutions in the exterior domain of a ball B e ( 0 ) with the center at the origin and the radius e > 0 when the ball shrink to the origin, that is, e → 0 + , for any fixed total mass h > 0 defined in (3.1), and then let h → 0 + .