Boundary layer

Physics

Trajectory

Work (physics)

Uniqueness

Similarity solution

Mathematics

Manifold (fluid mechanics)

Mechanics

Image (mathematics)

Similarity (geometry)

Astronomy

Vortex

Engineering

Mathematical analysis

Philosophy

Artificial intelligence

Computer science

Linguistics

Weak solution

Zero (linguistics)

Navier–Stokes equations

Compressibility

Mechanical engineering

Thermodynamics

Annals of Mathematics

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 Šverák (2015). Our solutions live precisely on the borderline of the known well-posedness theory.

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