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Difference between revisions of "Degenerate hyperbolic equation"

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A partial differential equation
 
A partial differential equation
  
<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/d/d030/d030790/d0307901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
F ( t, x, Du )  = 0
 +
$$
 +
 
 +
where the function  $  F( t, x, q) $
 +
satisfies the following condition: The roots of the polynomial
 +
 
 +
$$
 +
\sum _ {| \alpha | = m }
 +
 
 +
\frac{\partial  F ( t, x, Du ) }{\partial  q _  \alpha  }
  
where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d0307902.png" /> satisfies the following condition: The roots of the polynomial
+
\lambda ^ {\alpha _ {0} } \xi ^ {\alpha  ^  \prime  }
 +
$$
  
<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/d/d030/d030790/d0307903.png" /></td> </tr></table>
+
are real for all real  $  \xi $,
 +
and there exist  $  \xi \neq 0 $,
 +
$  t $,
 +
$  x $,
 +
and  $  Du $
 +
for which some of the roots either coincide or the coefficient of  $  \lambda  ^ {m} $
 +
vanishes. Here  $  t $
 +
is an independent variable which is often interpreted as time;  $  x $
 +
is an  $  n $-
 +
dimensional vector  $  ( x _ {1} \dots x _ {n} ) $;  
 +
$  u ( t, x) $
 +
is the unknown function; $  \alpha $
 +
and  $  \alpha  ^  \prime  $
 +
are multi-indices,  $  \alpha = ( \alpha _ {0} \dots \alpha _ {n} ) $,
 +
$  \alpha  ^  \prime  = ( \alpha _ {1} \dots \alpha _ {n} ) $;
 +
$  Du $
 +
is a vector with components
  
are real for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d0307904.png" />, and there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d0307905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d0307906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d0307907.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d0307908.png" /> for which some of the roots either coincide or the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d0307909.png" /> vanishes. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079010.png" /> is an independent variable which is often interpreted as time; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079011.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079012.png" />-dimensional vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079013.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079014.png" /> is the unknown function; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079016.png" /> are multi-indices, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079018.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079019.png" /> is a vector with components
+
$$
 +
D  ^  \alpha  u  = \
  
<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/d/d030/d030790/d03079020.png" /></td> </tr></table>
+
\frac{\partial  ^ {| \alpha | } u }{\partial  t ^ {\alpha _ {0} }
 +
\partial  x _ {1} ^ {\alpha _ {1} } \dots
 +
\partial  x _ {n} ^ {\alpha _ {n} } }
 +
;
 +
$$
  
only derivatives of an order not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079021.png" /> enter in equation (*); the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079022.png" /> are the components of a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079023.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079024.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079025.png" />-dimensional vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079026.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030790/d03079027.png" />.
+
only derivatives of an order not exceeding $  m $
 +
enter in equation (*); the $  q _  \alpha  $
 +
are the components of a vector $  q $;  
 +
$  \xi $
 +
is an $  n $-
 +
dimensional vector $  ( \xi _ {1} \dots \xi _ {n} ) $;  
 +
and $  \xi ^ {\alpha  ^  \prime  } = \xi _ {1} ^ {\alpha _ {1} } \dots \xi _ {n} ^ {\alpha _ {n} } $.
  
 
See also [[Degenerate partial differential equation|Degenerate partial differential equation]] and the references given there.
 
See also [[Degenerate partial differential equation|Degenerate partial differential equation]] and the references given there.

Latest revision as of 17:32, 5 June 2020


A partial differential equation

$$ \tag{* } F ( t, x, Du ) = 0 $$

where the function $ F( t, x, q) $ satisfies the following condition: The roots of the polynomial

$$ \sum _ {| \alpha | = m } \frac{\partial F ( t, x, Du ) }{\partial q _ \alpha } \lambda ^ {\alpha _ {0} } \xi ^ {\alpha ^ \prime } $$

are real for all real $ \xi $, and there exist $ \xi \neq 0 $, $ t $, $ x $, and $ Du $ for which some of the roots either coincide or the coefficient of $ \lambda ^ {m} $ vanishes. Here $ t $ is an independent variable which is often interpreted as time; $ x $ is an $ n $- dimensional vector $ ( x _ {1} \dots x _ {n} ) $; $ u ( t, x) $ is the unknown function; $ \alpha $ and $ \alpha ^ \prime $ are multi-indices, $ \alpha = ( \alpha _ {0} \dots \alpha _ {n} ) $, $ \alpha ^ \prime = ( \alpha _ {1} \dots \alpha _ {n} ) $; $ Du $ is a vector with components

$$ D ^ \alpha u = \ \frac{\partial ^ {| \alpha | } u }{\partial t ^ {\alpha _ {0} } \partial x _ {1} ^ {\alpha _ {1} } \dots \partial x _ {n} ^ {\alpha _ {n} } } ; $$

only derivatives of an order not exceeding $ m $ enter in equation (*); the $ q _ \alpha $ are the components of a vector $ q $; $ \xi $ is an $ n $- dimensional vector $ ( \xi _ {1} \dots \xi _ {n} ) $; and $ \xi ^ {\alpha ^ \prime } = \xi _ {1} ^ {\alpha _ {1} } \dots \xi _ {n} ^ {\alpha _ {n} } $.

See also Degenerate partial differential equation and the references given there.

How to Cite This Entry:
Degenerate hyperbolic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degenerate_hyperbolic_equation&oldid=46611
This article was adapted from an original article by A.M. Il'in (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article