An Example of Finite-time Singularities in the 3d Euler Equations

Let be the exterior of the closed unit ball. Consider the self-similar Euler system Setting α = β = 1/2 gives the limiting case of Leray’s self-similar Navier–Stokes equations. Assuming smoothness and smallness of the boundary data on ∂Ω, we prove that this system has a unique solution , vanishing at infinity, precisely The self-similarity transformation is v(x, t) = u(y)/(t* − t)^α, y = x/(t* − t)^β, where v(x, t) is a solution to the Euler equations. The existence of smooth function u(y) implies that the solution v(x, t) blows up at (x*, t*), x* = 0, t* < + ∞. This isolated singularity has bounded energy with unbounded L ² − norm of curl v.