Perelman's nonexistence conjecture for rotated backward self-similar Navier–Stokes profiles

Let $\alpha\neq 0$, and let $U\colon\mathbb{R}^3\to\mathbb{R}^3$ be a smooth solution of the profile system

$$\alpha\big(JU-(Jy\cdot\nabla)U\big)+\tfrac12U+\tfrac12(y\cdot\nabla)U-\Delta U+(U\cdot\nabla)U+\nabla P=0,$$ $$\nabla\cdot U=0$$

on $\mathbb{R}^3$. If there exists a constant $C_{U,0}>0$ such that

$$|U(y)|\leq\frac{C_{U,0}}{1+|y|}$$

for all $y\in\mathbb{R}^3$, then $U\equiv 0$.

Perelman's conjecture. Every smooth rotated backward self-similar profile with $\alpha\neq 0$ and the stated decay is trivial.

This conjecture concerns the exclusion of nontrivial rotated backward self-similar singularities for the three-dimensional incompressible Navier–Stokes equations. The corresponding non-rotated case is known to be ruled out under appropriate decay assumptions, whereas the rotated case remains open.