Cauchy-Kowalewski theorem

Consider a system of partial differential equationsMathworldPlanetmath involving m dependent variables u1,,um and n+1 independent variables t,x1,,xn:


in which F1,,Fm are analytic functions in a neighborhood of a point (u10,,um0;t0,x10,,xn0) subject to the boundary conditionsMathworldPlanetmath


when t=t0 for given functionsMathworldPlanetmath fi which are analytic in a neighborhood of x10,,xn0 such that ui0=fi(x10,,xn0).

The Cauchy-Kowalewski theorem asserts that this boundary value problem has a unique analytic solution ui=fi(t,x1,,xn) in a neighborhood of (u10,,um0;0,x10,,xn0).

The Cauchy-Kowalewski theorem is a local existence theoremMathworldPlanetmath — it only asserts that a solution exists in a neighborhood of the point, not in all space. A peculiar feature of this theorem is that the type of the differential equation (whether it is elliptic, parabolic, or hyperbolic) is irrelevant. As soon as one either considers global solutions or relaxes the assumptionPlanetmathPlanetmath of analyticity, this is no longer the case — the existence and uniqueness of solutions to a differential equation (or system of differential equations) will depend upon the type of the equation.

By simple transformations, one can generalize this theorem.

By making a change of variable t=t-ϕ(x1,,xm), with ϕ analytic, one can generalize the theorem to the case where the boundary values are specified on a surface given by the equation t=ϕ(x1,,xm) rather than on the plane t=t0.

One can allow higher order derivatives by the device of introducing new variables. For instance, to allow third order time derivatives of u1, one could introduce new variables u1t and u1tt and augment the system of equations by adding




Likewise, one can eliminate spatial derivativesPlanetmathPlanetmath. The manner in which one introduces new equations for these new variables is somewhat clumsy to describe in general, and it is best to explain it by example, as is done in a to this entry.

Title Cauchy-Kowalewski theorem
Canonical name CauchyKowalewskiTheorem
Date of creation 2013-03-22 14:37:04
Last modified on 2013-03-22 14:37:04
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 16
Author rspuzio (6075)
Entry type Theorem
Classification msc 35A10
Synonym Cauchy-Kovalevskaya theorem
Related topic Analytic
Related topic CauchyInitialValueProblem
Related topic ExistenceAndUniquenessOfSolutionOfOrdinaryDifferentialEquations