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

Abstract

In a seminal work, Leray (1934) 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 (2018). The second solution is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and \v Sverák (2015). Our solutions live precisely on the borderline of the known well-posedness theory.

Authors

Dallas Albritton

School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA

Elia Brué

School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA

Maria Colombo

Institute of Mathematics, EPFL SB, Lausanne, Switzerland