Nonexistence of self-similar singularities for the 3D incompressible Euler equations

We prove that there exists no self- similar finite time blowing up solution to the 3D incompressible Euler equations if the vorticity decays sufficiently fast near infinity in R-3. By a similar method we also show nonexistence of self- similar blowing up solutions to the divergence- free transport equation in R-n. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi- geostrophic equations, for which we also show nonexistence of self- similar blowing up solutions.