Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations

Abstract

We investigate homothetic forward self-similar solutions of the incompressible Navier-Stokes equations: the solutions $\overline{U}$ for which both $\overline{U}$ and $β\overline{U}$ are self-similar profiles for some nontrivial $β$. Homothetic solutions are, in addition, the only solutions for which a singular limit argument can be used to prove non-uniqueness of Leray-Hopf solutions along the lines of the Jia-Šverák program. In three dimensions, and for sufficiently regular initial data, we prove a Liouville theorem that rules out the existence of non-trivial homothetic solutions. In two dimensions, our Liouville theorem proves that the only decaying homothetic solution is the Oseen vortex. In addition, we prove that the Euler operator linearized around the Oseen vortex is stable. On the other hand, we also discover a homothetic solution for which an unstable approximate eigenvalue of the linearized Euler operator exists.