On rotated backwards self-similar solutions of the incompressible 3D Navier-Stokes equations

Abstract

We consider backwards globally self-similar solutions of the 3D incompressible Navier-Stokes equations which are invariant under the joint action of scaling (the natural parabolic scaling) and rotation about a given axis, at a constant angular speed $α$ in self-similar time. For these so-called rotated self-similar solutions (RSS), we prove that if they satisfy a Type~I upper bound, and if the rotation parameter $α$ is either too small, or too large, then they must be trivial. This Liouville-type result extends the classical works of Nečas-Růžička-Šverák ('96) and Tsai ('98), which only consider $α=0$, to the case of similarity profiles which experience nontrivial rotation. Our results partially answer a question posed by Perelman. For backwards globally self-similar solutions which are invariant under the discrete action of scaling and rotation, the so-called rotated discretely self-similar solutions (RDSS), we obtain similar Liouville-type results under a Type~I upper bound, assuming extreme values of the rotation parameter $α$, and if the scaling factor $λ$ is sufficiently close to $1$. The proof of all these results rests on the introduction of a robust weighted-$L^2$ framework. In particular, our method is quantitative and is not sensitive to whether the Bernoulli head pressure satisfies a maximum principle, which was a key obstruction in previous works.