Cauchy-Kovalevskaya theorem

A theorem stating that the Cauchy problem has a (unique) analytic solution locally if the functions occurring in the differential equation (or system of differential equations) and all the initial data, together with the non-characteristic initial surface, are analytic.

For a system of $ k $ partial differential equations in $ k $ unknown functions $ u _ {1} (x, x _ {0} ) \dots u _ {k} (x, x _ {0} ) $, of the form

$$ \tag{1 } \frac{\partial ^ {m} u _ {i} }{\partial x _ {0} ^ {m} } = \ F _ {i} \left ( x _ {0} , x, u ,\ \frac{\partial ^ {m _ {1} + \dots m _ {n} } u }{\partial x _ {0} ^ {m _ {0} } \dots \partial x _ {n} ^ {m _ {n} } } \right ) , $$

where

$$ i = 1 \dots k,\ \ x = (x _ {1} \dots x _ {n} ),\ \ u = (u _ {1} \dots u _ {k} ), $$

$$ \sum _ {j = 0 } ^ { n } m _ {j} \leq m,\ m _ {0} < m,\ m \geq 1, $$

the Cauchy–Kovalevskaya theorem states the following: The Cauchy problem posed by the initial data

$$ \tag{2 } \left . { \frac{\partial ^ {j} u _ {i} }{\partial x _ {0} ^ {j} } } \right | _ \sigma = \phi _ {ij} (x), \ i= 1 \dots k; \ j= 0 \dots m-1, $$

where $ \sigma = \{ {(x, x _ {0} ) } : {x _ {0} = 0, x \in \Omega _ {0} } \} $ is the initial surface of the data $ \phi _ {ij} $, has a unique analytic solution $ u (x, x _ {0} ) $ in some domain $ \Omega $ in $ (x _ {0} , x) $- space containing $ \Omega _ {0} \times \{ x _ {0} \} $, if $ F _ {i} $ and $ \phi _ {ij} $ are analytic functions of all their arguments.

Consider a linear system of differential equations

$$ \tag{3 } P (x, D) u \equiv \ \sum _ {| \alpha | \leq m } A _ \alpha (x) D ^ \alpha u = \ B (x), $$

where $ \alpha = ( \alpha _ {0} \dots \alpha _ {n} ) $ is a vector with non-negative integer coordinates;

$$ | \alpha | = \ \sum _ {j = 0 } ^ { n } \alpha _ {j} $$

is the order of the differential operator

$$ D ^ \alpha = \ D _ {0} ^ {\alpha _ {0} } \dots D _ {n} ^ {\alpha _ {n} } ,\ \ D _ {j} = \ \frac \partial {\partial x _ {j} } ,\ \ j = 0 \dots n; $$

$ A _ \alpha (x) $, $ x = (x _ {0} \dots x _ {n} ) $, is a given square matrix of order $ N $; $ u (x) = \| u _ {j} (x) \| $, $ j = 1 \dots N $, is the column-vector of unknown functions; and $ B (x) $ is a given $ N $- vector.

Generally speaking, the Cauchy–Kovalevskaya theorem does not exclude the possibility of the Cauchy problem having non-analytic solutions in addition to the analytic solution. However, the Cauchy problem for a linear system (3) with analytic coefficients $ A _ \alpha (x) $ and Cauchy conditions on an analytic non-characteristic surface $ \sigma $ has at most one solution in a certain neighbourhood $ \Omega _ {0} $ of $ \sigma $. This assertion is valid without any analyticity assumptions concerning the initial data and the solution $ u (x) $.

A solution of the Cauchy problem (1), (2), the existence of which is guaranteed by the Cauchy–Kovalevskaya theorem, may turn out to be unstable (since a small variation of the initial data $ \phi _ {ij} (x) $ may induce a large variation of the solution). For example, this is the case when the system (1) is of elliptic type. If the initial data are non-analytic, the Cauchy problem (1), (2) may become meaningless unless attention is confined to hyperbolic systems (1).

For a broad class of equations, the Cauchy–Kovalevskaya theorem may be generalized to the case in which the initial manifold is characteristic at every point (see [1], [2]). In that case the initial functions cannot be prescribed at will; they must satisfy certain conditions, dictated by the differential equation.

A characteristic Cauchy problem (cf. Cauchy characteristic problem) may have more than one solution. In particular, one has the following proposition. Let $ P (x, D) $ be a differential equation of order $ m $ with principal part $ P _ {m} (x, D) $ and with real-analytic coefficients, defined in a neighbourhood $ \Omega $ of a point $ x ^ {0} $ of the Euclidean space $ \mathbf R ^ {n} $, and let $ \phi $ be a real-analytic function in $ \Omega $ such that

$$ \mathop{\rm grad} \phi (x ^ {0} ) \neq 0 \ \ \textrm{ and } \ \ P _ {m} (x, \mathop{\rm grad} \phi ) = 0, $$

but $ P _ {m} ^ {(j)} (x, \mathop{\rm grad} \phi ) \neq 0 $ at $ x = x ^ {0} $ for some $ j $. Then there exists a neighbourhood $ \Omega _ {0} $ of $ x ^ {0} $ and a function $ u (x) $ of class $ C ^ {m} ( \Omega _ {0} ) $, analytic whenever $ \phi (x) \neq \phi (x ^ {0} ) $, such that $ P (x, D) u = 0 $ and

$$ \supp u = \ \{ {x } : {x \in \Omega _ {0} ,\ \phi (x) \leq \phi (x ^ {0} ) } \} . $$

If the initial manifold is characteristic along certain curves then, generally speaking, the solution of the characteristic Cauchy problem is non-unique in some neighbourhood of the initial surface and the degree of bifurcation is defined by the geometrical nature of the corresponding characteristic surfaces. The theorem was first proved by S.V. Kovalevskaya (1875).

References

[1]	L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)
[2]	A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) (Translated from Russian)
[3]	V.S. Vladimirov, "Equations of mathematical physics" , MIR (1984) (Translated from Russian)
[4]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[5]	L. Hörmander, "Linear partial differential operators" , Springer (1963)

Comments

The uniqueness result in the case of non-analytic data is Holmgren's theorem (see [5], Part II Chapt. 5, or [a2], Theorem 8.6.5). A modern proof of the Cauchy–Kovalevskaya theorem in the linear case can be found in [a2], Sect. 9.4. A relatively short proof can be found in [a1].

The theorem is also called the Cauchy–Kovalevski or Cauchy–Kowalewsky theorem.

References

[a1]	J. Chazarain, A. Piriou, "Introduction to the theory of linear partial differential equations" , North-Holland (1982) (Translated from French)
[a2]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983)