Navier–Stokes existence and smoothness problem
E175568
The Navier–Stokes existence and smoothness problem is a fundamental unsolved question in mathematical fluid dynamics that asks whether three-dimensional fluid flow equations always have smooth, globally defined solutions.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Navier–Stokes existence and smoothness problem canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T1523331 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Navier–Stokes existence and smoothness problem Context triple: [Millennium Prize Problem, includes, Navier–Stokes existence and smoothness problem]
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A.
Yang–Mills existence and mass gap problem
The Yang–Mills existence and mass gap problem is a fundamental unsolved question in mathematical physics that asks for a rigorous proof that quantum Yang–Mills theory exists and exhibits a positive mass gap, and is one of the seven Millennium Prize Problems.
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B.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
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C.
Navier–Stokes equations
The Navier–Stokes equations are fundamental partial differential equations in fluid mechanics that describe how the velocity field of a fluid evolves under forces like pressure and viscosity.
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D.
Millennium Prize Problem
The Millennium Prize Problem is one of seven famous unsolved mathematical problems designated by the Clay Mathematics Institute, each carrying a $1 million reward for a correct solution.
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E.
P versus NP problem
The P versus NP problem is a central unsolved question in theoretical computer science that asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Navier–Stokes existence and smoothness problem Target entity description: The Navier–Stokes existence and smoothness problem is a fundamental unsolved question in mathematical fluid dynamics that asks whether three-dimensional fluid flow equations always have smooth, globally defined solutions.
-
A.
Yang–Mills existence and mass gap problem
The Yang–Mills existence and mass gap problem is a fundamental unsolved question in mathematical physics that asks for a rigorous proof that quantum Yang–Mills theory exists and exhibits a positive mass gap, and is one of the seven Millennium Prize Problems.
-
B.
Poincaré conjecture
The Poincaré conjecture is a landmark problem in topology that characterizes the three-dimensional sphere among three-dimensional manifolds and was famously solved by Grigori Perelman in the early 2000s.
-
C.
Navier–Stokes equations
The Navier–Stokes equations are fundamental partial differential equations in fluid mechanics that describe how the velocity field of a fluid evolves under forces like pressure and viscosity.
-
D.
Millennium Prize Problem
The Millennium Prize Problem is one of seven famous unsolved mathematical problems designated by the Clay Mathematics Institute, each carrying a $1 million reward for a correct solution.
-
E.
P versus NP problem
The P versus NP problem is a central unsolved question in theoretical computer science that asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Millennium Prize Problem
ⓘ
open problem in mathematics ⓘ problem in partial differential equations ⓘ |
| asksWhether |
finite-time singularities can occur
ⓘ
solutions exist for all time ⓘ solutions remain smooth for all time ⓘ |
| boundaryCondition | either whole space or periodic boundary conditions ⓘ |
| concerns | Navier–Stokes equations ⓘ |
| dataType |
finite-energy initial data
ⓘ
smooth initial data ⓘ |
| difficultySource |
lack of a priori bounds preventing blow-up
ⓘ
possible energy cascade to small scales ⓘ |
| dimension | three-dimensional case ⓘ |
| domain |
three-dimensional Euclidean space
ⓘ
three-dimensional periodic box ⓘ |
| equationProperty |
incompressibility condition div u = 0
ⓘ
viscosity term given by Laplacian of velocity ⓘ |
| equationType |
incompressible Navier–Stokes equations
ⓘ
viscous fluid equations ⓘ |
| field |
analysis of partial differential equations
ⓘ
mathematical fluid dynamics ⓘ mathematical physics ⓘ |
| hasFormulation |
existence and smoothness over R^3 with given external force
ⓘ
uniqueness of smooth solutions given smooth initial data ⓘ |
| hasPrize | 1 million US dollars ⓘ |
| implication |
would advance understanding of nonlinear PDEs
ⓘ
would clarify mathematical foundations of turbulence ⓘ would impact numerical simulation of fluids ⓘ |
| knownResult |
global smooth solutions exist in two dimensions
ⓘ
global weak solutions exist in three dimensions (Leray solutions) ⓘ local-in-time smooth solutions exist in three dimensions ⓘ |
| listedIn |
Millennium Prize Problem
ⓘ
surface form:
Clay Millennium Problems
|
| namedAfter |
Claude-Louis Navier
ⓘ
George Stokes ⓘ
surface form:
George Gabriel Stokes
|
| regularityQuestion |
absence of blow-up in finite time
ⓘ
global regularity of solutions ⓘ |
| relatedConcept |
blow-up criterion
ⓘ
energy inequality ⓘ strong solution ⓘ turbulence ⓘ weak solution ⓘ |
| relatedEquationProperty |
elliptic-parabolic character of the system
ⓘ
nonlinearity of the convective term ⓘ |
| sponsoredBy | Clay Mathematics Institute ⓘ |
| status | unsolved ⓘ |
| timeInterval | all positive times ⓘ |
| variableType |
pressure field of a fluid
ⓘ
velocity field of a fluid ⓘ |
| yearFormulatedAsMillenniumProblem | 2000 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Navier–Stokes existence and smoothness problem Description of subject: The Navier–Stokes existence and smoothness problem is a fundamental unsolved question in mathematical fluid dynamics that asks whether three-dimensional fluid flow equations always have smooth, globally defined solutions.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Mémoire sur les lois du mouvement des fluides
→
associatedWith
→
Navier–Stokes existence and smoothness problem
ⓘ