# Cauchy–Kowalevski theorem

In mathematics, the **Cauchy–Kowalevski theorem** (also written as the **Cauchy–Kovalevskaya theorem**) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proven by Augustin Cauchy (1842), and the full result by Sophie Kovalevskaya (1875).

## Contents

## First order Cauchy–Kowalevski theoremEdit

This theorem is about the existence of solutions to a system of *m* differential equations in *n* dimensions when the coefficients are analytic functions. The theorem and its proof are valid for analytic functions of either real or complex variables.

Let *K* denote either the fields of real or complex numbers, and let *V* = *K*^{m} and *W* = *K*^{n}. Let *A*_{1}, ..., *A*_{n−1} be analytic functions defined on some neighbourhood of (0, 0) in *W* × *V* and taking values in the *m* × *m* matrices, and let *b* be an analytic function with values in *V* defined on the same neighbourhood. Then there is a neighbourhood of 0 in *W* on which the quasilinear Cauchy problem

with initial condition

on the hypersurface

has a unique analytic solution *ƒ* : *W* → *V* near 0.

Lewy's example shows that the theorem is not valid for all smooth functions.

The theorem can also be stated in abstract (real or complex) vector spaces. Let *V* and *W* be finite-dimensional real or complex vector spaces, with *n* = dim *W*. Let *A*_{1}, ..., *A*_{n−1} be analytic functions with values in End (*V*) and *b* an analytic function with values in *V*, defined on some neighbourhood of (0, 0) in *W* × *V*. In this case, the same result holds.

## Proof by analytic majorizationEdit

Both sides of the partial differential equation can be expanded as formal power series and give recurrence relations for the coefficients of the formal power series for *f* that uniquely determine the coefficients. The Taylor series coefficients of the *A*_{i}'s and *b* are majorized in matrix and vector norm by a simple scalar rational analytic function. The corresponding scalar Cauchy problem involving this function instead of the *A*_{i}'s and *b* has an explicit local analytic solution. The absolute values of its coefficients majorize the norms of those of the original problem; so the formal power series solution must converge
where the scalar solution converges.

## Higher-order Cauchy–Kowalevski theoremEdit

If *F* and *f*_{j} are analytic functions near 0, then the non-linear Cauchy problem

with initial conditions

has a unique analytic solution near 0.

This follows from the first order problem by considering the derivatives of *h* appearing on the right hand side as components of a vector-valued function.

### ExampleEdit

The heat equation

with the condition

has a unique formal power series solution (expanded around (0, 0)). However this formal power series does not converge for any non-zero values of *t*, so there are no analytic solutions in a neighborhood of the origin. This shows that the condition |*α*| + *j* ≤ *k* above cannot be dropped. (This example is due to Kowalevski.)

## Cauchy–Kowalevski–Kashiwara theoremEdit

There is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients, the Cauchy–Kowalevski–Kashiwara theorem, due to Masaki Kashiwara (1983). This theorem involves a cohomological formulation, presented in the language of D-modules. The existence condition involves a compatibility condition among the non homogeneous parts of each equation and the vanishing of a derived functor .

### ExampleEdit

Let . Set . The system has a solution if and only if the compatibility conditions are verified. In order to have a unique solution we must include an initial condition , where .

## ReferencesEdit

- Cauchy, Augustin (1842), "Mémoire sur l'emploi du calcul des limites dans l'intégration des équations aux dérivées partielles",
*Comptes rendus*,**15**Reprinted in Oeuvres completes, 1 serie, Tome VII, pages 17–58. - Folland, Gerald B. (1995),
*Introduction to Partial Differential Equations*, Princeton University Press, ISBN 0-691-04361-2 - Hörmander, L. (1983),
*The analysis of linear partial differential operators I*, Grundl. Math. Wissenschaft.,**256**, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035 (linear case) - Kashiwara, M. (1983),
*Systems of microdifferential equations*, Progress in Mathematics,**34**, Birkhäuser, ISBN 0817631380 - von Kowalevsky, Sophie (1875), "Zur Theorie der partiellen Differentialgleichung",
*Journal für die reine und angewandte Mathematik*,**80**: 1–32 (German spelling of her surname used at that time.) - Nakhushev, A.M. (2001) [1994], "Cauchy–Kovalevskaya theorem", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4