Calculus of Variations and Geometric Measure Theory

D. Albritton - E. Bruè - M. Colombo

Non-uniqueness of Leray solutions of the forced Navier-Stokes equations

created by bruè on 30 Dec 2021
modified on 11 Mar 2022


Accepted Paper

Inserted: 30 dec 2021
Last Updated: 11 mar 2022

Journal: Annals of Mathematics
Year: 2021


In the seminal work 39, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. We exhibit two distinct Leray solutions with zero initial velocity and identical body force. Our approach is to construct a `background' solution which is unstable for the Navier-Stokes dynamics in similarity variables; its similarity profile is a smooth, compactly supported vortex ring whose cross-section is a modification of the unstable two-dimensional vortex constructed by Vishik in 43,44. The second solution is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Šverák in 32,33. Our solutions live precisely on the borderline of the known well-posedness theory.