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By definition, the perfect alignment between the vorticity vector and the vortex stretching one means that there is the same perfect alignment between the vorticity vector and the strain matrix eigenvector. We prove the following. For the “restricted Euler equations”, if there is a time instant such that the aforesaid perfect vector alignment happens at some point inside the flow, it continues then permanently and keeps forever in the sense that one happens successively at points which belongs to the given point trajectory generated by the flow. Further, if such a trajectory exists, depending on initial data, one can either “blow up” for a finite time or not. The aforementioned blows up can be very different. What is interesting in doing so is that there is such kind the finite-time blows up that eigenvalues of the strain matrix all go at infinity which can be as positive as negative whereas the vorticity remains bounded at the same time. This is, generally speaking, unexpected if to take into account the well-known BKM criterion of finite-time blows up for solutions to the Euler equations. As for the (original) Euler equations, if their solution is analytic, and: (1) there is a finite while (let very small even) such that the perfect vector alignment happens at both some point inside the flow and points belonging to the given point trajectory fraction which is generated by the flow during the course of the given while, then it continues permanently and keeps forever in the same sense as above; (2) there is no such while and the perfect vector alignment happens at a time instant only, a question respecting of the subsequent perfect vector alignment existence remains open.

Perfect vector alignment of the vorticity and the vortex stretching is one forever