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. The proof uses the vorticity transport formula represented in terms of the back to label map. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in R. 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. 1 Incompressible Euler equations We are concerned here on the following Euler equations for the homogeneous incompressible fluid flows in R. (E)    ∂v ∂t + (v · ∇)v = −∇p, (x, t) ∈ R × (0,∞) div v = 0, (x, t) ∈ R × (0,∞) v(x, 0) = v0(x), x ∈ R 3 ∗The work was supported partially by the KOSEF Grant no. R01-2005-000-10077-0.