Liouville conjecture for stationary Navier–Stokes fields

Let $v$ be a vector field on $\mathbb R^{3}$, with pressure $p$, satisfying the stationary Navier–Stokes system

$$-\nu \Delta v+\sum_{1\le j\le 3}\partial_{j}(v_{j}v)+\nabla p=0,\qquad \operatorname{div}v=0,$$

where $\nu>0$, together with

$$v\in L^{1}_{\operatorname{loc}}(\mathbb R^{3},\mathbb R^{3})\cap\mathscr S'(\mathbb R^{3},\mathbb R^{3}),\qquad \forall s>0,\ \left|\{x\in\mathbb R^{3}:|v(x)|>s\}\right|<+\infty,$$

and $\operatorname{curl}v\in L^{2}(\mathbb R^{3},\mathbb R^{3})$. Liouville conjecture. Then $v$ is identically equal to $0$. This is the paper’s main conjecture and asserts that sufficiently decaying stationary three-dimensional incompressible Navier–Stokes fields must vanish; the source states that it remains open.