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

Abstract.

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

$$ \alpha U + \beta y \cdot \nabla U + U \cdot \nabla U + \nabla P = 0,\quad {\text{div }}U = 0. $$

For $\alpha = \beta = 1/2,$ which is the limiting case of Leray’s self-similar Navier–Stokes equations, we prove the existence of $(U,P) \in H^3 (\Omega ,\mathbb{R}^3 \times \mathbb{R})$ in a smooth bounded domain Ω, with the inflow boundary data of non-zero vorticity. This implies the possibility that solutions of the Euler equations blow up at a time $t = t^* ,t^* < + \infty .$