Self-similar solutions of stationary Navier-Stokes equations

In this paper, we mainly study the existence of self-similar solutions of stationary Navier-Stokes equations for dimension n=3,4. For n=3, if the external force is axisymmetric, scaling invariant, C1,α continuous away from the origin and small enough on the sphere S2, we shall prove that there exists a family of axisymmetric self-similar solutions which can be arbitrarily large in the class Cloc3,α(R3\0). Moreover, for axisymmetric external forces without swirl, corresponding to this family, the momentum flux of the flow along the symmetry axis can take any real number. However, there are no regular (U∈Cloc3,α(R3\0)) axisymmetric self-similar solutions provided that the external force is a large multiple of some scaling invariant axisymmetric F which cannot be driven by a potential. In the case of dimension 4, there always exists at least one self-similar solution to the stationary Navier-Stokes equations with any scaling invariant external force in L4/3,∞(R4).