Cauchy problem

E239288

initial value problem mathematical concept problem in partial differential equations

The Cauchy problem is a fundamental type of initial value problem in partial differential equations, where one seeks a solution satisfying prescribed data on a given initial surface.

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All labels observed (3)

Label	Occurrences
Cauchy problem canonical	4
Cauchy problem in differential equations	1
Hadamard well-posedness	1

How this entity was disambiguated

This entity first appeared as the object of triple T2171649 — 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: Cauchy problem
Context triple: [Augustin-Louis Cauchy, knownFor, Cauchy problem]

A. Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
B. d’Alembert’s formula
d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
C. ODE
ODE is the state agency responsible for overseeing public education and implementing education policy in Oregon.
D. Hamilton–Jacobi equation
The Hamilton–Jacobi equation is a fundamental partial differential equation in classical mechanics that reformulates dynamics in terms of a generating function, providing a powerful bridge to quantum mechanics and modern analytical methods.
E. local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
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: Cauchy problem
Target entity description: The Cauchy problem is a fundamental type of initial value problem in partial differential equations, where one seeks a solution satisfying prescribed data on a given initial surface.

A. Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
B. d’Alembert’s formula
d’Alembert’s formula is a classical solution method for the one-dimensional wave equation that expresses the displacement of a vibrating string in terms of its initial shape and velocity.
C. ODE
ODE is the state agency responsible for overseeing public education and implementing education policy in Oregon.
D. Hamilton–Jacobi equation
The Hamilton–Jacobi equation is a fundamental partial differential equation in classical mechanics that reformulates dynamics in terms of a generating function, providing a powerful bridge to quantum mechanics and modern analytical methods.
E. local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	initial value problem ⓘ mathematical concept ⓘ problem in partial differential equations ⓘ
appliesTo	ordinary differential equations ⓘ partial differential equations ⓘ
concerns	propagation of information from initial data ⓘ
coreIdea	determine future and past evolution from initial state ⓘ
difficulty	may fail to be well-posed for elliptic equations ⓘ
example	Schrödinger equation with initial wave function ⓘ heat equation with initial temperature distribution ⓘ wave equation with initial displacement and velocity ⓘ
field	applied mathematics ⓘ mathematical analysis ⓘ partial differential equations ⓘ
hasComponent	differential equation ⓘ initial data ⓘ initial surface ⓘ
hasDefinition	problem of finding a solution of a differential equation that satisfies prescribed initial data on a given set ⓘ
hasProperty	may be well-posed or ill-posed ⓘ
historicalPeriod	19th century ⓘ
namedAfter	Augustin-Louis Cauchy ⓘ
relatedTo	Cauchy–Kowalevski theorem ⓘ linked to: Cauchy–Kovalevskaya theorem Cauchy–Lipschitz theorem ⓘ linked to: local existence and uniqueness theorem boundary value problem ⓘ elliptic partial differential equation ⓘ existence and uniqueness theorem ⓘ hyperbolic partial differential equation ⓘ parabolic partial differential equation ⓘ well-posed problem ⓘ
requires	specification of data on a non-characteristic surface for hyperbolic PDEs ⓘ
solvedBy	Fourier transform methods ⓘ energy methods ⓘ fixed point theorems ⓘ semigroup theory ⓘ
specialCaseOf	initial value problem ⓘ
studiedIn	Sobolev spaces ⓘ functional analysis ⓘ microlocal analysis ⓘ theory of distributions ⓘ
typicalFormulation	find u solving F(x,u,Du,…) = 0 with u\|_S = f on an initial surface S ⓘ
usedIn	control theory ⓘ engineering ⓘ fluid dynamics ⓘ general relativity ⓘ mathematical physics ⓘ
wellPosednessCriteria	continuous dependence on initial data ⓘ existence of solution ⓘ uniqueness of solution ⓘ

How these facts were elicited

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Augustin-Louis Cauchy → knownFor → Cauchy problem ⓘ

Jacques Hadamard → knownFor → Hadamard well-posedness ⓘ

linked to: Cauchy problem

Lax equivalence theorem → relatedTo → Cauchy problem ⓘ

Peano existence theorem → relatedConcept → Cauchy problem ⓘ

Cauchy–Kovalevskaya theorem → relatedTo → Cauchy problem ⓘ

Augustin-Louis Cauchy → notableFor → Cauchy problem in differential equations ⓘ

subject linked to: Augustin-Louis