Cauchy–Kovalevskaya theorem

In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski 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 Sofya Kovalevskaya (1874).

First order Cauchy–Kovalevskaya theorem

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 the field of real or of complex numbers, and let V = Km and W = Kn. Let A1, ..., An−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

∂

x

n

f

=

A

1

(

x

,

f

)

∂

x

1

f

+

⋯

+

A

n

−

1

(

x

,

f

)

∂

x

n

−

1

f

+

b

(

x

,

f

)

{\displaystyle \partial _{x_{n}}f=A_{1}(x,f)\partial _{x_{1}}f+\cdots +A_{n-1}(x,f)\partial _{x_{n-1}}f+b(x,f)}

with initial condition

f

(

x

)

=

0

{\displaystyle f(x)=0}

on the hypersurface

x

n

=

0

{\displaystyle x_{n}=0}

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

Lewy's example shows that the theorem is not more generally 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 A1, ..., An−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.