# Stationary Navier-Stokes solutions

Are there known nontrivial ($$u\neq0$$) stationary solutions to Navier-Stokes equations in $$\mathbb R^3$$ ? Not square integrable of course (that's impossible), but with self-similar amplitudes of Fourier modes such as $$|\hat u(c\xi)|=c^{-2}|\hat u(\xi)|$$ ?

That would be $$\Delta u=B(u,u)$$ with $$\nabla\cdot u=0$$, where $$B(u,v):=\sum_{i=1}^3\partial_i(u_iv)+\nabla p$$ and $$p$$ is such that $$\nabla\cdot B(u,v)=0$$.

Same question for the particular class of cylindrically symmetric solutions. In cylindrical coordinates: $$(r,\theta,z)$$ and $$(u_r,u_\theta,u_z,p)$$ not depending on $$\theta$$, with $$\frac1r\frac\partial{\partial r}(ru_r)+\frac{\partial u_z}{\partial z}=0$$ (incompressibility).

I wonder if a solution with $$u(c\mathbb x)=c^{-1}u(\mathbb x)$$ is possible...

Tai-Peng Tsai's book Lectures on Navier-Stokes Equations (2018) cites as Theorem 8.3 (p.149) a theorem of V. Sverak (2011) that excludes the existence of minus one homogeneous solutions on $$\mathbb R^3$$. Technically, the only such solutions on $$\mathbb R^3-\{0\}$$ are the so-called Landau or Slezkin-Landau solutions, which in $$\mathbb R^3$$ satisfy the stationary Navier-Stokes equation with an exterior force $$\mathbb b\delta$$ (and vanish if $$\mathbb b=0$$).
What remains from the question is the possible existence of solutions with some other scaling, at infinity only, like $$u\propto U(\alpha,\theta)r^{-2/3}$$ in spherical coordinates. Such a scaling allows kinetic energy input from infinitely large scales, that would compensate viscous dissipation at small scales.