nLab Navier-Stokes equation

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Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id ⊣ id ∨ ∨ fermionic ⇉ ⊣ ⇝ bosonic ⊥ ⊥ bosonic ⇝ ⊣ Rh rheonomic ∨ ∨ reduced ℜ ⊣ ℑ infinitesimal ⊥ ⊥ infinitesimal ℑ ⊣ & étale ∨ ∨ cohesive ʃ ⊣ ♭ discrete ⊥ ⊥ discrete ♭ ⊣ ♯ continuous ∨ ∨ ∅ ⊣ * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Equality and Equivalence

equivalence

principle of equivalence

equation

Contents

Idea

The partial differential equation describing viscous hydrodynamics.

References

See also

Discussion of the (relativistic) Navier-Stokes equation in view of the fluid/gravity correspondence:

An AI-generated proof, verified with Lean, of instability of the NS-equations in the presence of a smooth external force:

  • OpenAI math: On the Navier–Stokes Millennium Prize Problem (8 Sep 2026) [openai.com/index/navier-stokes-solution]

  • Tristan Buckmaster: Blowup for the Euler equations with smooth forcing, talk at Mathematics in the Age of AI, Courant NYU (24 Sep 2026) [video:yt, nodes:pdf]

That OpenAI’s techniques do not extend to the NS-equations in the presence of a real analytic external force:

  • Peter Constantin, Mihaela Ignatovam, Vlad Vicol: Regularity of asymptotically axisymmetric solutions to the 3D Navier–Stokes equations with analytic forcing (17 September 2026) [arXiv:2609.20803]

Last revised on October 7, 2026 at 05:34:17. See the history of this page for a list of all contributions to it.

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