Global energy-level observability for nonlinear three-dimensional Navier–Stokes equations

Establish a global observability inequality at the energy level for the fully nonlinear three-dimensional incompressible Navier–Stokes equations that is compatible with Leray–Hopf weak solutions, with uniform constants and without additional regularity assumptions.

Background

The paper develops observability and coercivity results for the linear damped Stokes system, but explicitly distinguishes these from the nonlinear three-dimensional Navier–Stokes problem. A global energy-level observability inequality for the nonlinear system would provide a direct way to relate localized dissipation to global control for Leray–Hopf weak solutions. The paper states that such an inequality is currently unavailable, so the nonlinear analysis uses localized coercivity only as an assumed structural property and derives consequences through the energy inequality.