An example of finite-time singularities in the 3d Euler equations (Q2460212)

The author studies the blow-up of the solution to three-dimensional Euler equation. Employing the self-similarity transformation of Leray, he transforms the Euler equation to a self-similar system: \[ \alpha u +\beta x\cdot \nabla u +u\cdot \nabla u +\nabla p =0,\quad \operatorname{div} u=0, \] for some constants \(\alpha ,\beta\). For this self-similar system, he gives some boundedness results in an annular domain \(\Omega \subset \mathbb{R}^3\), prescribing a boundary condition on \(\partial\Omega\). Using this result, the author develops an existence and uniqueness theory for the self-similar system. This enables one to construct a solution to the Euler equation, which has an isolated singularity at \(t=t^* \in (0,\infty )\), \(x=x^* \in \mathbb{R}^3\). For this solution, the rotation blows up, but the energy does not.