Theorem

Cauchy-Kovalevskaya Theorem

Analysis

The Cauchy-Kovalevskaya Theorem guarantees the existence and local uniqueness of a solution to a partial differential equation together with initial data prescribed on a noncharacteristic surface, provided every function involved is analytic. Named for Augustin-Louis Cauchy and Sofya Kovalevskaya, it is the principal general existence theorem for analytic partial differential equations, though its analyticity requirement makes it inapplicable to many equations of physical interest that are studied instead by other, more specialized methods.

Facts

Statement

For a system of partial differential equations whose coefficients and initial data are all given by analytic functions, and in which the equations can be solved for the highest order derivative transverse to the initial surface, there exists a unique analytic solution in a neighborhood of that surface. 1

Proof Year

1874 1

Sofya Kovalevskaya proved the full theorem in 1874, published in 1875. Augustin Cauchy had proved a special case in 1842.

Classification

Statement Form

Existence Theorem 1

Statement Form

Uniqueness Theorem 1

Connections

Proved By

Augustin-Louis Cauchy

Sources

1. Cauchy-Kovalevskaya Theorem (Wikipedia)

Wikimedia FoundationLead section, second sentence

Quote, Lead section, second sentence

A special case was proven by Augustin Cauchy (1842), and the full result by Sofya Kovalevskaya (1874).

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Cauchy-Kovalevskaya Theorem

Proved By