Let alpha≠ 0, and let U colonR^3→R^3 be a smooth solution of the profile system alpha big(JU-(Jy·nabla)U big)+tfrac12U+tfrac12(y·nabla)U-Delta U+(U·nabla)U+nabla P=0, nabla· U=0 on R^3. If there exists a constant C_U,0>0 such that |U(y)|≤fracC_U,01+|y| for all y∈R^3, then U equiv 0. Perelman's conjecture. Every smooth rotated backward self-similar profile with alpha≠ 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.

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.

Perelman’s conjecture, attributed to G. Perelman through private communication to G. Seregin, asserts that every smooth rotated profile with the stated decay is zero for every $\alpha\neq 0$ . The non-rotated case is known, but the full rotated problem remains open.

Known results

For $\alpha=0$ , nonexistence under suitable decay was proved by Nečas, Růžička, and Šverák, and by Tsai.
For $\alpha\neq 0$ , the borderline decay gives only weak- $L^3$ control, and rotation obstructs the earlier maximum-principle method.

2026 partial theorem

A 2026 arXiv preprint claims, using a weighted- $L^2$ enstrophy argument, that the profile is trivial when either $|\alpha|<\underline{\alpha}(C_{U,0})$ or $|\alpha|>\overline{\alpha}(C_{U,0})$ . Thus sufficiently small and sufficiently large rotation parameters are covered, while the intermediate regime, including $\alpha\approx 1$ , remains open.

Current status (as of September 2026): The conjecture is claimed for sufficiently small and sufficiently large $|\alpha|$ , while the intermediate range remains open; this partial result is unverified.

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

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Solutions 0