Cauchy–Kovalevskaya theorem

The Cauchy–Kovalevskaya theorem guarantees a local analytic solution to certain partial differential equations when the initial data and coefficients are analytic and the initial hypersurface is noncharacteristic.

Cauchy problem: An initial-value problem that seeks solutions to a differential equation from prescribed data on an initial surface. The theorem establishes local solvability for a restricted class of these initial-value problems.
Partial differential equation: An equation involving an unknown function of several variables and its partial derivatives. The theorem concerns local solutions to this broad class of equations.
Cauchy–Kovalevskaya theorem for smooth data: There is no general extension of the Cauchy–Kovalevskaya existence guarantee from analytic data to arbitrary smooth data. Smoothness alone does not provide the convergent power-series control required by the theorem.
Augustin-Louis Cauchy: A French mathematician whose work shaped analysis, complex function theory, and differential equations. The theorem bears Cauchy's name for his role in its early development.
Analytic function: A function that is locally represented by a convergent power series. Analyticity of the coefficients and initial data lets the theorem construct a convergent solution series.
Initial value problem: A differential equation together with conditions specifying a solution at an initial point or surface. Analytic data on a hypersurface supply the initial conditions in the theorem.
Hadamard's example: A classical example of a well-posed elliptic equation whose Cauchy problem is unstable. It demonstrates that analytic local solvability need not imply stable dependence on initial data.
Sofya Kovalevskaya: A Russian mathematician known for work on differential equations, analysis, and rigid-body motion. Her 1875 work established the general theorem now associated with her name.
Noncharacteristic surface: A hypersurface on which the principal symbol of a differential operator does not vanish in the normal direction. This condition allows the equation to determine successive normal derivatives from the initial data.
Local existence theorem: A result proving that a differential equation has a solution near specified initial data. The theorem guarantees existence only in a neighborhood of the initial hypersurface.

Linked from 2 pages

Existence and uniqueness theoremCompared with: It extends the existence-and-uniqueness theme to a restricted class of partial differential equations.