- 1. How many settings are there for the Navier–Stokes existence and smoothness problem described in the text?
A) Two
B) Three
C) Four
D) One
- 2. What is the alternative to the Euclidean space setting to avoid problems at infinity?
A) Spherical coordinates
B) Periodic framework
C) Manifold projection
D) Bounded domains
- 3. What is the 3-dimensional torus represented as in the text?
A) R3 / Z3
B) R3 / R3
C) Z3 / R3
D) T3 / Z3
- 4. What is assumed about the initial condition v0(x) in Euclidean space?
A) It is incompressible and fractal
B) It is constant and zero
C) It is smooth and divergence-free
D) It is periodic and discontinuous
- 5. How is the external force f(x,t) categorized in the Euclidean version of the problem?
A) Gaussian, white noise, or constant
B) Periodic, impulsive, or infinite
C) Smooth, real analytic, or equal to zero
D) Linear, quadratic, or static
- 6. What does condition 1 represent in the Navier–Stokes Euclidean setting?
A) Non-zero pressure gradients
B) Smoothness and global definition
C) Conservation of mass
D) Boundary conditions
- 7. What does condition 2 represent in the Navier–Stokes Euclidean setting?
A) Zero divergence
B) Infinite spatial decay
C) Constant velocity field
D) Globally bounded kinetic energy
- 8. In the existence and smoothness statement, what is the external force assumed to be when considering the initial condition?
A) f(x,t) = constant
B) f(x,t) = 1
C) f(x,t) is unknown
D) f(x,t) = 0
- 9. What is the mathematical space defined for the velocity and pressure functions?
A) C1(R3 x [0, infinity))
B) L2(R3 x [0, infinity))
C) Cinfinity(R3 x [0, infinity))
D) Linfinity(R3 x [0, infinity))
- 10. What are the characteristics of the initial condition and external force in the breakdown setting?
A) Smoothness, analyticity, or zero
B) Discontinuity, periodicity, or growth
C) Constant, linear, or decaying
D) Bounded, periodic, or divergent
- 11. What kind of equations are the Navier–Stokes equations?
A) Nonlinear partial differential equations
B) Algebraic equations
C) Linear ordinary differential equations
D) Stochastic differential equations
- 12. In the context of the Navier–Stokes equations, what does the symbol ν (nu) represent?
A) External force
B) Kinematic viscosity
C) Density
D) Pressure
- 13. Which physical law do the Navier–Stokes equations state?
A) The ideal gas law
B) Conservation of angular momentum
C) Conservation of energy
D) Newton's second law
- 14. What is the additional equation used to solve for the pressure in an incompressible fluid?
A) The energy equation
B) The continuity equation
C) The vorticity equation
D) The state equation
- 15. In 2000, which organization named the Navier–Stokes existence and smoothness problem as one of seven Millennium Prize Problems?
A) The Fields Institute
B) Clay Mathematics Institute
C) OpenAI
D) The Royal Society
- 16. What term is used to describe the property of being divergence-free in the context of Navier–Stokes solutions?
A) Laminar
B) Compressible
C) Solenoidal
D) Inviscid
- 17. Which company claimed to have found a solution to the Navier–Stokes equations with a smooth external force in 2026?
A) OpenAI
B) DeepMind
C) Microsoft
D) Anthropic
- 18. Which mathematical operator is represented by Δ or ∇²?
A) Divergence
B) Gradient
C) Curl
D) Laplacian
- 19. What term is used to describe the convective term (v · ∇)v in the Navier–Stokes equations?
A) Stochastic
B) Linear
C) Nonlinear
D) Constant
- 20. Who provided the official specific statements of the Millennium Prize problem for Navier–Stokes?
A) Tristan Buckmaster
B) Diego Córdoba
C) Levent Alpöge
D) Charles Fefferman
- 21. How many total instances of the Navier–Stokes problem were named as Millennium Prize Problems?
A) Seven
B) Four
C) Two
D) One
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