Finite-energy Liouville theorem for three-dimensional stationary Navier–Stokes equations

Prove that every solution of the three-dimensional stationary Navier–Stokes equations on \(\mathbb{R}^3\) satisfying \(u\in \dot H^1(\mathbb{R}^3;\mathbb{R}^3)\) and \(u(x)\to0\) as \(|x|\to\infty\) is identically zero.

Background

The paper places its two-dimensional fractional Liouville theorem in the broader context of stationary Navier–Stokes rigidity results. In three dimensions, the authors note that the corresponding conclusion under the natural finite-Dirichlet-energy condition and pointwise decay remains unresolved. Existing results establish triviality only after imposing additional assumptions, such as membership in an auxiliary Lebesgue, BMO-based, or related function space. Thus, the open problem is to determine whether the stated H˙1\dot H^1 energy and decay hypotheses alone exclude all nontrivial stationary solutions in three dimensions.

References

It is remarked in Remark X.9.4 and that it remains an open problem to prove that a solution of the three-dimensional stationary Navier--Stokes equations satisfying

u \in \Hdot{1}(\mathbb{R}3;R3) \qquad\text{and}\qquad u(x) \to 0 \quad \text{as } |x| \to +\infty

must be identically zero.

— A Liouville theorem for the two-dimensional stationary hypodissipative Navier--Stokes system  (2608.16437 - Nitti et al., 17 Aug 2026) in Section 1, subsection “Survey of the literature”