Navier-Stokes Singularity Problem

A Navier-Stokes Singularity Problem is a millennium prize problem that asks whether smooth solutions to the Navier-Stokes equations can develop singularities in finite time from smooth initial conditions.

• AKA: Navier-Stokes Existence and Smoothness Problem, NS Blowup Problem, Navier-Stokes Regularity Problem.
• Context:
  • It can typically involve Solution Regularity in three-dimensional space.
  • It can typically question Finite-Time Blowup from smooth initial conditions.
  • It can typically challenge Mathematical Analysis of nonlinear PDEs.
  • It can often motivate Numerical Simulations for singularity evidence.
  • It can often inspire Theoretical Developments in fluid dynamics.
  • It can often connect to Turbulence Theory through energy cascades.
  • It can range from being a 2D Navier-Stokes Singularity Problem to being a 3D Navier-Stokes Singularity Problem, depending on its spatial dimension.
  • It can range from being a Viscous Navier-Stokes Singularity Problem to being an Inviscid Navier-Stokes Singularity Problem, depending on its viscosity presence.
  • It can range from being a Periodic Navier-Stokes Singularity Problem to being a Whole-Space Navier-Stokes Singularity Problem, depending on its domain topology.
  • It can range from being a Deterministic Navier-Stokes Singularity Problem to being a Stochastic Navier-Stokes Singularity Problem, depending on its noise inclusion.
  • ...
• Example:
  • Related Singularity Problems, such as:
    • Euler Equation Singularity Problem for inviscid fluids.
    • MHD Singularity Problem for magnetohydrodynamic flows.
  • Partial Results, such as:
    • Caffarelli-Kohn-Nirenberg Result on singularity measure.
    • Leray's Weak Solution for existence theory.
  • ...
• Counter-Example:
  • 2D Navier-Stokes Problem, which has global regularity proven.
  • Linear Heat Equation Problem, which admits smooth solutions globally.
• See: Millennium Prize Problems, Navier-Stokes Equation, Mathematical Singularity, Partial Differential Equation, GR-Navier-Stokes Linking Method, Fluid Dynamics, Clay Mathematics Institute.

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