Numerical Investigations of Non-uniqueness for the Navier-Stokes Initial Value Problem in Borderline Spaces

We consider the Cauchy problem for the incompressible Navier-Stokes equations in R-3 for a one-parameter familyof explicit scale-invariant axi-symmetric initial data, which is smooth away from the origin and invariant under the re?ectionwith respect to the xy-plane. Working in the class of axi-symmetric ?elds, we calculate numerically scale-invariant solutionsof the Cauchy problem in terms of their pro?le functions, which are smooth. The solutions are necessarily unique for smalldata, but for large data we observe a breaking of the re?ection symmetry of the initial data through a pitchfork-typebifurcation. By a variation of previous results by Jia and ?Sver'ak (Invent Math 196(1):233-265, 2013, https://doi.org/10.1007/s00222-013- 0468-x) it is known rigorously that if the behavior seen here numerically can be proved, optimal non-uniqueness examples for the Cauchy problem can be established, and two di?erent solutions can exists for the same initialdatum which is divergence-free, smooth away from the origin, compactly supported, and locally (-1)-homogeneous nearthe origin. In particular, assuming our (?nite-dimensional) numerics represents faithfully the behavior of the full (in?nite-dimensional) system, the problem of uniqueness of the Leray-Hopf solutions (with non-smooth initial data) has a negativeanswer and, in addition, the perturbative arguments such those by Kato (Math Z 187(4):471-480, 1984, https://doi.org/10.1007/BF01174182) and Koch and Tataru (Adv Math 157(1):22-35, 2001, https://doi.org/10.1006/aima.2000.1937), orthe weak-strong uniqueness results by Leray, Prodi, Serrin, Ladyzhenskaya and others, already give essentially optimalresults. There are no singularities involved in the numerics, as we work only with smooth pro?le functions. It is conceivablethat our calculations could be upgraded to a computer-assisted proof, although this would involve a substantial amount ofadditional work and calculations, including a much more detailed analysis of the asymptotic expansions of the solutions atlarge distances.