Sverak’s Landau rigidity conjecture in three dimensions

Prove that every solution of the stationary three-dimensional Navier–Stokes equations on \(\mathbb{R}^3\setminus\{0\}\) satisfying \(|u(x)|\le C/|x|\) is a Landau solution, without any smallness assumption on \(C\).

Background

The paper discusses the classification of singular stationary Navier–Stokes solutions with the critical decay bound ∣u(x)∣=O(∣x∣−1)|u(x)|=O(|x|^{-1}). Landau solutions are known to be rigid under axisymmetric self-similar and, by earlier work, general self-similar restrictions. The unresolved problem is whether the same classification remains valid when only the critical pointwise bound is imposed in three dimensions, particularly for large values of the bound constant CC. The results of the paper establish rigidity only in restricted rotated self-similar, discretely self-similar, and rotated discretely self-similar regimes.

References

But if we further relax the restriction to only eq:scale_bound, the rigidity of Landau solution is still unknown and conjuctured by \v{S}verak in : Each solution of eq:sns in \mathbb{R}3\backslash{0}$ satisfies |u(x)|\le\frac{C}{|x|} must be a Landau solution.

eq:scale_bound:

∣u(x)∣≤C∣x∣|u(x)|\le\frac{C}{|x|}

eq:sns:

{ −Δu+u⋅∇u+∇p=0 div⁡u=0in R3\{0}\left\{\begin{aligned} &\,-\Delta u+u\cdot\nabla u+\nabla p=0\\ &\qquad\quad\ \operatorname{div}u=0 \end{aligned}\right.\quad\text{in } \mathbb{R}^3\backslash\{0\}

— Rigidity of Landau solution in the rotated self-similar class  (2609.24261 - Shi et al., 21 Sep 2026) in Section 1, subsection “Rigidity problem for steady Navier Stokes equations and main result”

Without smallness, even the identification of the leading order term remains open, and a general Landau rigidity result will provide this through a blow down argument, as in the higher dimensional setting .

— Rigidity of Landau solution in the rotated self-similar class  (2609.24261 - Shi et al., 21 Sep 2026) in Section 1, subsection “Motivation of rigidity result from asymptotic behavior and regularity”

Hence in , Tsai mentioned a conjecture that if |u|\le C/|x| or equivalently u\in L{3,\infty} without smallness, whether u is regular?

— Rigidity of Landau solution in the rotated self-similar class  (2609.24261 - Shi et al., 21 Sep 2026) in Section 1, subsection “Motivation of rigidity result from asymptotic behavior and regularity”