3
$\begingroup$

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...

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.