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.

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