Sofia Kovalevskaya

Leigh Ellison

Cauchy-Kovalevskaya Theorem

Whilst working under the supervision of Weierstrass, Sofia was to write three papers which he felt were of the necessary standard for a doctoral degree. The first and most highly acclaimed of these dealt with the theory of partial differential equations. Weierstrass had previously worked on theorems involving ordinary differential equations and wished to extend some of them to partial differential equations. It has been said by Cooke that,

Apparently he thought that a general theorem could be proved to the effect that a power series obtained formally from a partial differential equation in which only analytic functions occur would necessarily converge.

It was this conjecture that Sofia was supposed to try to prove. Her work included the formulation of what is commonly known as the Cauchy-Kovalevskaya Theorem, and it was said by Henri Poincaré that her work, significantly simplified Cauchy's method of proof, and gave the theorem its final form.

It was in 1872 that Cauchy presented the first general existence theorem in partial differential equations which showed that ordinary differential equations and some partial differential equations have analytic solutions. If we look at an analytic function in a neighbourhood of the point (t, u) which is subject to

$f(t,u) = \large \frac {du}{dt}$

with the initial conditions u = u₀ and t = t₀ , then there exists a unique solution. Cauchy looked at partial differential equations such as

${ \large \frac {\partial \mu }{ \partial t}}\normalsize = F_{i}(t, x_{1} , ... , x_{n}: u_{1} , ... , u_{n}: \large \frac {\partial u_{1}}{\partial x_{1}} , ... , \frac {\partial u_{m}}{\partial x_{m}}\normalsize)$

with the initial conditions u₁(0, x₁ , ... , x_n) = w_i(x₁ , ... , x_n) for i = 1, ... , m when t = 0, and tried to find a solution which satisfied these conditions. This he did by finding a locally convergent power series solution having assumed that F_i and w_i were analytic. A simple analytic function then replaces F_i which has non-negative coefficients in the power series expansion that are greater than or equal to the absolute value of the corresponding coefficients for F_i.

It was this result that Sofia was to generalise in her dissertation to systems of order r with time derivatives ∂^rμ/∂t^r. One of the more subtle forms of the Cauchy-Kovalevskaya Theorem tells us that an equation ∂u/∂t = f(t, x, u, ∂μ/∂x) where f is analytic for values near (t₀ , x₀ , u₀ , ∂μ/∂x) has exactly one solution u(x, t) which is analytic near (x₀ , t₀). In proving this theorem Sofia was unsurprisingly influenced by Cauchy's use of majorants in his work to obtain locally convergent power series solutions, and used majorants of the form

$M/{1 - [(t_{1} + t_{2} + ... + t_{r})/\rho]}$

where

\rho

and M are constraints.

This theorem is somewhat restricted by the fact that it is limited to analytic functions and may fail in certain regions, but nevertheless plays a very basic role in the study of differential equations.