The 3D Navier-Stokes Problem (original) (raw)

Abstract

It is not known whether the three-dimensional (3D) incompressible Navier-Stokes equations possess unique smooth (continuously differentiable) solutions at high Reynolds numbers. This problem is quite important for basic science, practical applications, and numerical computations. This review presents a selective survey of the current state of the mathematical theory, focusing on the technical source of difficulties encountered with the construction of smooth solutions.

Key takeaways

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  1. The existence of unique smooth solutions to the 3D Navier-Stokes equations remains unresolved at high Reynolds numbers.
  2. The review discusses mathematical challenges surrounding solution existence, uniqueness, and regularity for expert audiences.
  3. Weak solutions exist but may lack uniqueness, complicating the analysis of turbulent flows.
  4. Vortex stretching significantly impacts the mathematical difficulties in establishing regularity of solutions.
  5. A $1 million prize incentivizes resolution of fundamental questions within the Navier-Stokes problem.

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References (38)

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FAQs

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What explains the necessity for a mathematical foundation for the Navier-Stokes equations?add

The study reveals that a firm mathematical foundation ensures predictive robustness, requiring solutions to exist, be unique, and continuously depend on initial data, particularly for turbulent flow conditions prevalent in various scientific fields.

How does vortex stretching impact the regularity of solutions in 3D Navier-Stokes equations?add

Vortex stretching leads to increased enstrophy, complicating the regularity of solutions and contributing to uncertainties regarding finite-time singularities, making vortex dynamics a critical area of study.

What role do weak solutions play in understanding the Navier-Stokes equations?add

Weak solutions permit analysis under less stringent smoothness conditions, allowing proofs of existence; however, their uniqueness remains an unresolved issue, particularly in turbulent regimes.

When were the primary mathematical challenges of the Navier-Stokes equations highlighted?add

The review underscores significant unresolved mathematical questions first noted over seventy years ago, emphasizing ongoing complexities regarding existence, uniqueness, and regularity of solutions.

What are the implications of Reynolds number on solution uniqueness in Navier-Stokes equations?add

The analysis indicates that uniqueness and smoothness of solutions are generally assured only for very low Reynolds numbers, which precludes turbulence—a critical factor in fluid dynamics.