Global Regularity for the Navier-Stokes Equations via the Energy Contraction Operator

Key Points

• To prove the global existence of smooth solutions for the three-dimensional incompressible Navier-Stokes equations.
• Introduced the energy contraction operator to analyze the equations.
• Assumed a finite-time singularity scenario to assess energy behavior.
• Derived contradictions based on total kinetic energy conservation.
• Established that assuming a singularity leads to unbounded energy dissipation.
• Demonstrated that total kinetic energy conservation holds under smooth solutions.