Existence of decaying self-similar solutions to the three-dimensional Euler equations

Establish the existence of solutions to the stationary self-similar three-dimensional incompressible Euler equation with decay at spatial infinity.

Background

The paper studies stationary and self-similar solutions of the three-dimensional incompressible Euler equations under axisymmetry. The self-similar equation is substantially more difficult than the stationary vortex equation, and only a small number of examples with limited regularity are known.

The authors emphasize that constructing self-similar Euler solutions that decay at infinity remains a fundamental unresolved issue. Their contribution constructs a self-similar profile with an asymptotic self-similar decay, but the broader existence problem for decaying self-similar solutions is explicitly identified as open.

References

These results left open whether self-similar singular profiles persist for stronger convection but are not reached by the forward dynamics.

— Stable Singularity of the Euler Equations on $\mathbb{R}^3$  (2609.10867 - Ganeshram et al., 9 Sep 2026) in Section 1, Introduction; Appendix A, Section 'Convection-varied equations'

The existence of solutions to selfsim with decay at infinity is a major open problem in the field.

— On smooth inviscid vortices with fat tails  (2609.04786 - Raphaël et al., 4 Sep 2026) in Section 1, subsection “Previous constructions and related problems,” paragraph “Self similar solutions”