Existence of Singular Self-Similar Solutions of the Three-Dimensional Euler Equations in a Bounded Domain

A self-similar solution of the three-dimensional (3d) incompressible Euler equations is defined by u(x, t) = U(y)/(t* - t)(alpha), y = x/(t* - t)(beta), alpha,beta > 0, where U(y) satisfies alpha U + beta y . del U + U . del U + del P = 0, div U = 0. For alpha = beta = 1/2, which is the limiting case of Leray's self-similar Navier-Stokes equations, we prove the existence of (U, P) is an element of H-3 (Omega, R-3 x R) in a smooth bounded domain Omega, with the inflow boundary data of non-zero vorticity. This implies the possibility that solutions of the Euler equations blow up at a time t = t*, t* < +infinity.