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Difference between revisions of "Cauchy-Kovalevskaya theorem"

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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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c0209201.png" /> partial differential equations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c0209202.png" /> unknown functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c0209203.png" />, of the form
+
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} } }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c0209204.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\right ) ,
 +
$$
  
 
where
 
where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c0209205.png" /></td> </tr></table>
+
$$
 +
i = 1 \dots k,\ \
 +
x = (x _ {1} \dots x _ {n} ),\ \
 +
u = (u _ {1} \dots u _ {k} ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c0209206.png" /></td> </tr></table>
+
$$
 +
\sum _ {j = 0 } ^ { n }  m _ {j}  \leq  m,\  m _ {0} < m,\  m \geq  1,
 +
$$
  
 
the Cauchy–Kovalevskaya theorem states the following: The Cauchy problem posed by the initial data
 
the Cauchy–Kovalevskaya theorem states the following: The Cauchy problem posed by the initial data
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c0209207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\left . {
 +
\frac{\partial  ^ {j} u _ {i} }{\partial  x _ {0}  ^ {j} }
 +
}
 +
\right | _  \sigma  = \phi _ {ij} (x),
 +
\  i= 1 \dots k; \  j= 0 \dots m-1,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c0209208.png" /> is the initial surface of the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c0209209.png" />, has a unique analytic solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092010.png" /> in some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092012.png" />-space containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092013.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092015.png" /> are analytic functions of all their arguments.
+
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
 
Consider a linear system of differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
P (x, D)
 +
u  \equiv \
 +
\sum _ {| \alpha | \leq  m }
 +
A _  \alpha  (x)
 +
D  ^  \alpha  u  = \
 +
B (x),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092017.png" /> is a vector with non-negative integer coordinates;
+
where $  \alpha = ( \alpha _ {0} \dots \alpha _ {n} ) $
 +
is a vector with non-negative integer coordinates;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092018.png" /></td> </tr></table>
+
$$
 +
| \alpha |  = \
 +
\sum _ {j = 0 } ^ { n }
 +
\alpha _ {j}  $$
  
 
is the order of the differential operator
 
is the order of the differential operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092019.png" /></td> </tr></table>
+
$$
 +
D  ^  \alpha  = \
 +
D _ {0} ^ {\alpha _ {0} } \dots
 +
D _ {n} ^ {\alpha _ {n} } ,\ \
 +
D _ {j}  = \
 +
 
 +
\frac \partial {\partial  x _ {j} }
 +
,\ \
 +
j = 0 \dots n;
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092021.png" />, is a given square matrix of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092022.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092024.png" />, is the column-vector of unknown functions; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092025.png" /> is a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092026.png" />-vector.
+
$  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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092027.png" /> and Cauchy conditions on an analytic non-characteristic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092028.png" /> has at most one solution in a certain neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092029.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092030.png" />. This assertion is valid without any analyticity assumptions concerning the initial data and the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092031.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092032.png" /> 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).
+
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 [[#References|[1]]], [[#References|[2]]]). In that case the initial functions cannot be prescribed at will; they must satisfy certain conditions, dictated by the differential equation.
 
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 [[#References|[1]]], [[#References|[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|Cauchy characteristic problem]]) may have more than one solution. In particular, one has the following proposition. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092033.png" /> be a differential equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092034.png" /> with principal part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092035.png" /> and with real-analytic coefficients, defined in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092036.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092037.png" /> of the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092038.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092039.png" /> be a real-analytic function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092040.png" /> such that
+
A characteristic Cauchy problem (cf. [[Cauchy characteristic problem|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092041.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm grad}  \phi (x  ^ {0} )  \neq  0 \ \
 +
\textrm{ and } \ \
 +
P _ {m} (x,  \mathop{\rm grad}  \phi )  = 0,
 +
$$
  
but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092042.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092043.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092044.png" />. Then there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092046.png" /> and a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092047.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092048.png" />, analytic whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092049.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092050.png" /> and
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020920/c02092051.png" /></td> </tr></table>
+
$$
 +
\supp  u  = \
 +
\{ {x } : {x \in \Omega _ {0} ,\
 +
\phi (x) \leq  \phi (x  ^ {0} ) } \}
 +
.
 +
$$
  
 
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).
 
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).
Line 49: Line 153:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bers,  F. John,  M. Schechter,  "Partial differential equations" , Interscience  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Hörmander,  "Linear partial differential operators" , Springer  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bers,  F. John,  M. Schechter,  "Partial differential equations" , Interscience  (1964)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Bitsadze,  "Equations of mathematical physics" , MIR  (1980)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.S. Vladimirov,  "Equations of mathematical physics" , MIR  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial differential equations" , '''2''' , Interscience  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Hörmander,  "Linear partial differential operators" , Springer  (1963)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 15:35, 4 June 2020


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)
How to Cite This Entry:
Cauchy-Kovalevskaya theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy-Kovalevskaya_theorem&oldid=22261
This article was adapted from an original article by A.M. Nakhushev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article