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

@misc{binz2026homothetic,
  title = {{Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations}},
  author = {Tim Binz and Matei P. Coiculescu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.12159},
  eprint = {2607.12159},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.12159},
}

TY  - UNPB
TI  - Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations
AU  - Tim Binz
AU  - Matei P. Coiculescu
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2607.12159
UR  - https://arxiv.org/abs/2607.12159
AB  - We investigate homothetic forward self-similar solutions of the incompressible Navier-Stokes equations: the solutions U̅ for which both U̅ and βU̅ 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(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.
ER  -