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

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–Kovalevskaya theorem ⓘ surface form: Cauchy–Kowalevski theorem local existence and uniqueness theorem ⓘ surface form: Cauchy–Lipschitz 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

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: Cauchy problem
Description of subject: 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.

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 → Cauchy problem ⓘ

this entity surface form: Hadamard well-posedness

Lax equivalence theorem → relatedTo → Cauchy problem ⓘ

Peano existence theorem → relatedConcept → Cauchy problem ⓘ

Cauchy–Kovalevskaya theorem → relatedTo → Cauchy problem ⓘ

Augustin-Louis → notableFor → Cauchy problem ⓘ

subject surface form: Augustin-Louis Cauchy

this entity surface form: Cauchy problem in differential equations