Global energy identity for 3D weak Navier–Stokes solutions

Establish whether weak (Leray–Hopf) solutions of the three-dimensional incompressible Navier–Stokes equations satisfy the exact global kinetic energy balance with equality (the energy identity), rather than only the global energy inequality, for all positive times and suitable initial data.

Background

The paper discusses the role of the global energy identity in connecting statistical descriptions of turbulent flows to Kolmogorov’s theory. While Leray–Hopf solutions are known to satisfy only the energy inequality, the equality is a key assumption in deriving Kolmogorov’s power laws for self-similar homogeneous statistical solutions.

Demonstrating the exact energy identity for weak solutions in three dimensions would resolve a long-standing issue and strengthen the mathematical foundations underlying statistical approaches to turbulence.