Year: 2026
Type: preprint
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 reproves a result of Luo and Shvydkoy that the only decaying homothetic solution with continuous angular data is the Oseen vortex. In addition, we prove that the Euler equations linearized around the Oseen vortex are stable in some subspaces of $L^2(\mathbb{R}^2)$, for example in the space of $2$-fold rotationally symmetric functions. On the other hand, we also discover a homothetic solution for which an unstable approximate eigenvalue of the linearized Euler equations exists.
Source: arXiv (Cornell University)
Authors: Binz, Tim, Coiculescu, Matei P.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7168341702