Existence of the axisymmetric weak solution to the 3D isothermal stationary compressible Navier-Stokes equations

In this paper, we construct the axisymmetric weak solutions to the 3D isothermal stationary compressible Navier-Stokes equations on the domain D, where the heat ratio equal 1 and the external force g satisfies certain cancellation conditions. We first establish the compactness assertion of the approximation solutions by assuming the total mass of the fluid is finite, then we exclude the trivial solution via imposing another type of restrictions on the density. To deal with the singularities near the symmetric axis and at the far field, we will derive proper weighted estimates of the L2-norm of the velocity field on the unbounded domain D, which belongs to the critical case of the Sobolev's inequality.