math.AP — Analysis of PDEs

Non-Uniqueness of Leray-Hopf Solutions to the Unforced Three-Dimensional Navier-Stokes Equations

Contributed by Tristan Buckmaster, Matei Coiculescu, Chen Li

We prove that the incompressible Navier--Stokes equations in $\R^3$, without external force, admit a continuum of distinct suitable weak solutions in the Leray--Hopf class with the same initial datum. The datum is compactly supported, axisymmetric without swirl, and smooth away from the origin; it lies in $L^{3,\infty}$ and in $L^q$ for every $q<3$, but not in $L^3$. The solutions are axisymmetric without swirl and smooth at positive times, and they satisfy the local and global energy equalities. Any two of them separate in $L^2$ at the rate $t^{1/4+\sigma}$, for a fixed $0<\sigma<1/4$. The proof follows the program of Jia and \v Sver\'ak, with a discretely self-similar background in place of a self-similar one. The background is an exact unforced solution built from a train of thin vortex rings, which we obtain by lifting a periodic row of planar vortices with Gaussian stream function. We construct a growing Floquet mode of its linearization directly, from the angular-two instability of the planar vortex, and conclude by a nonlinear correction and a localization of the datum.

Author comments: Lean formalization forthcoming at the public repository at https://github.com/mcoiculescu/unforced-leray-hopf-public Exposition will be improved in future versions. We used an adaptation of Scott Armstrong's skill files for the Lean formalization process.