Stationary and discontinuous weak solutions of the Navier-Stokes equations

Abstract

We prove that there exists a nontrivial finite energy periodic stationary weak solution to the 3D Navier-Stokes equations (NSE). The construction relies on a convex integration scheme utilizing new stationary building blocks designed specifically for the NSE. The constructed family of approximate stationary solutions is also used to prove the existence of weak solutions of the NSE with energy profiles discontinuous on a dense set of positive Lebesgue measure.

Unfortunately, a critical error was found in the construction in Section 3 and the weak solutions constructed based on the building blocks in Section 3 were not divergence-free

References in corpus (2)

• The vanishing viscosity limit for a dyadic model
• Non-unique weak solutions in Leray-Hopf class of the 3D Hall-MHD system

Cited by in corpus (6)

• Convex integration solutions to the transport equation with full dimensional concentration
• Non-uniqueness of Weak Solutions to the 3D Quasi-Geostrophic Equations
• Nonuniqueness of weak solutions for the transport equation at critical space regularity
• Anomalous dissipation, anomalous work, and energy balance for smooth solutions of the Navier-Stokes equations
• Weak solutions of the three-dimensional hypoviscous elastodynamics with finite kinetic energy
• Non-uniqueness of weak solutions to 2D hypoviscous Navier-Stokes equations