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''at a given point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483001.png" />''
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A partial differential equation for which the [[Cauchy problem|Cauchy problem]] is uniquely solvable for initial data specified in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483002.png" /> on any non-characteristic surface (cf. [[Characteristic surface|Characteristic surface]]). In particular, a partial differential equation for which the normal cone has no imaginary zones is a hyperbolic partial differential equation. The differential equation
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/h/h048/h048300/h0483003.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
''at a given point  $  M( x _ {1} \dots x _ {n} ) $''
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483004.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483005.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483006.png" /> is a homogeneous polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483007.png" />, while the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483008.png" /> is of lower degree than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h0483009.png" />, is a hyperbolic partial differential equation if its characteristic equation
+
A partial differential equation for which the [[Cauchy problem|Cauchy problem]] is uniquely solvable for initial data specified in a neighbourhood of $  M $
 +
on any non-characteristic surface (cf. [[Characteristic surface|Characteristic surface]]). In particular, a partial differential equation for which the normal cone has no imaginary zones is a hyperbolic partial differential equation. The 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/h/h048/h048300/h04830010.png" /></td> </tr></table>
+
$$ \tag{* }
 +
L ( u)  = H ( D _ {1} \dots D _ {n} ) u +
 +
F ( D _ {1} \dots D _ {n} ) u + G ( x)  = 0,
 +
$$
  
has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830011.png" /> different real solutions with respect to one of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830012.png" />, the remaining ones being fixed. Any equation (*) of the first order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830013.png" /> with real coefficients is a hyperbolic partial differential equation. A second-order equation
+
where  $  D _ {i} = \partial  / \partial  x _ {i} $(
 +
$  i = 1 \dots n $),  
 +
$  H( D _ {1} \dots D _ {n} ) $
 +
is a homogeneous polynomial of degree  $  m $,
 +
while the polynomial  $  F $
 +
is of lower degree than  $  m $,
 +
is a hyperbolic partial differential equation if its characteristic 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/h/h048/h048300/h04830014.png" /></td> </tr></table>
+
$$
 +
Q ( \xi _ {1} \dots \xi _ {n} )  = H ( \xi _ {1} \dots \xi _ {n} )  = 0
 +
$$
 +
 
 +
has  $  m $
 +
different real solutions with respect to one of the variables  $  \xi _ {1} \dots \xi _ {n} $,
 +
the remaining ones being fixed. Any equation (*) of the first order  $  ( m = 1 ) $
 +
with real coefficients is a hyperbolic partial differential equation. A second-order equation
 +
 
 +
$$
 +
L ( u)  = u _ {tt} -
 +
\sum _ {i, j = 1 } ^ { n }
 +
a _ {ij} D _ {i} D _ {j} u + Fu + G  = 0
 +
$$
  
 
is hyperbolic if the quadratic form
 
is hyperbolic if the quadratic form
  
<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/h/h048/h048300/h04830015.png" /></td> </tr></table>
+
$$
 +
\sum _ {i, j = 1 } ^ { n }  a _ {ij} \xi _ {i} \xi _ {j}  $$
  
 
is positive definite.
 
is positive definite.
  
 +
====Comments====
 +
The special variable among the  $  \xi _ {1} \dots \xi _ {n} $
 +
such that  $  H ( \xi _ {1} \dots \xi _ {n} ) $
 +
has  $  m $
 +
different real solutions for each set of fixed values of the other  $  n - 1 $
 +
is often taken to be  $  t $(
 +
time). One speaks then of a (strictly) hyperbolic equation or an equation of (strictly) hyperbolic type with respect to the  $  t $-
 +
direction. More generally one considers hyperbolicity with respect to a vector  $  N $[[#References|[a1]]].
 +
 +
A polynomial  $  P $
 +
of degree  $  m $
 +
with principal part  $  P _ {m} $
 +
is called hyperbolic with respect to the real vector  $  N $
 +
if  $  P _ {m} ( N) \neq 0 $
 +
and there exists a number  $  \tau _ {0} > 0 $
 +
such that
 +
 +
$$
 +
P ( \xi + i \tau N)  \neq  0 \ \
 +
\textrm{ if } \
 +
\xi \in \mathbf R  ^ {n} ,\
 +
\tau < \tau _ {0} .
 +
$$
  
 +
If  $  P _ {m} $
 +
is such that  $  P _ {m} ( N) \neq 0 $
 +
and  $  P _ {m} ( \xi + \tau N) $
 +
has only simple real roots for every real  $  \xi \neq 0 $,
 +
then  $  P $
 +
is said to be strictly hyperbolic or hyperbolic in the sense of Petrovskii.
  
====Comments====
+
The Cauchy problem for a constant-coefficient differential operator  $  P $
The special variable among the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830016.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830017.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830018.png" /> different real solutions for each set of fixed values of the other <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830019.png" /> is often taken to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830020.png" /> (time). One speaks then of a (strictly) hyperbolic equation or an equation of (strictly) hyperbolic type with respect to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830021.png" />-direction. More generally one considers hyperbolicity with respect to a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830022.png" /> [[#References|[a1]]].
+
with data on a non-characteristic plane is well posed for arbitrary lower-order terms if and only if  $  P $
 +
is strictly hyperbolic. For a discussion of similar matters for polynomials  $  P $
 +
with variable coefficients cf. [[#References|[a2]]].
  
A polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830023.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830024.png" /> with principal part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830025.png" /> is called hyperbolic with respect to the real vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830027.png" /> and there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830028.png" /> such that
+
For a system of higher-order linear partial 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/h/h048/h048300/h04830029.png" /></td> </tr></table>
+
$$
 +
\sum _ {j = 1 } ^ { l }
 +
\sum _ {| \alpha | \leq  N _ {j} }
 +
a _  \alpha  ^ {ij} ( x)
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830030.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830032.png" /> has only simple real roots for every real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830033.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830034.png" /> is said to be strictly hyperbolic or hyperbolic in the sense of Petrovskii.
+
\frac{\partial  ^  \alpha  }{\partial  x  ^  \alpha  }
  
The Cauchy problem for a constant-coefficient differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830035.png" /> with data on a non-characteristic plane is well posed for arbitrary lower-order terms if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830036.png" /> is strictly hyperbolic. For a discussion of similar matters for polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830037.png" /> with variable coefficients cf. [[#References|[a2]]].
+
u _ {j}  = 0,\ \
 +
i = 1 \dots l ,
 +
$$
  
For a system of higher-order linear partial differential equations
+
where  $  \alpha = ( \alpha _ {0} \dots \alpha _ {n} ) $,
 +
is a hyperbolic system of partial differential equations in the sense of Petrovskii if the determinant
  
<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/h/h048/h048300/h04830038.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm det} \
 +
\left (
 +
\sum _ {| \alpha | \leq  N _ {j} }
 +
a _  \alpha  ^ {ij}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830039.png" />, is a hyperbolic system of partial differential equations in the sense of Petrovskii if the determinant
+
\frac{\partial   ^  \alpha  }{\partial  x  ^  \alpha  }
  
<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/h/h048/h048300/h04830040.png" /></td> </tr></table>
+
\right )
 +
$$
  
calculated in the ring of differential operators is a hyperbolic polynomial in the sense of Petrovskii (as a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830041.png" />). The Cauchy problem for a system that is hyperbolic in this sense is well posed [[#References|[a3]]], [[#References|[a4]]].
+
calculated in the ring of differential operators is a hyperbolic polynomial in the sense of Petrovskii (as a polynomial of degree $  N = \sum N _ {j} $).  
 +
The Cauchy problem for a system that is hyperbolic in this sense is well posed [[#References|[a3]]], [[#References|[a4]]].
  
Instead of strictly hyperbolic one also finds the term strongly hyperbolic and instead of hyperbolic also weakly hyperbolic (which is therefore the case in which the lower-order terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048300/h04830042.png" /> do matter).
+
Instead of strictly hyperbolic one also finds the term strongly hyperbolic and instead of hyperbolic also weakly hyperbolic (which is therefore the case in which the lower-order terms of $  P $
 +
do matter).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  pp. Chapt. XII</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''III''' , Springer  (1985)  pp. Chapt. XXIII</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.G. Petrovskii,  "Ueber das Cauchysche Problem für Systeme von partiellen Differentialgleichungen"  ''Mat. Sb. (N.S.)'' , '''2(44)'''  (1937)  pp. 815–870</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Mizohata,  "The theory of partial differential equations" , Cambridge Univ. Press  (1973)  (Translated from Japanese)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Chaillou,  "Hyperbolic differential polynomials" , Reidel  (1979)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Chazarain,  "Opérateurs hyperboliques à characteristique de multiplicité constante"  ''Ann. Inst. Fourier'' , '''24'''  (1974)  pp. 173–202</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Gårding,  "Linear hyperbolic equations with constant coefficients"  ''Acta Math.'' , '''85'''  (1951)  pp. 1–62</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  O.A. Oleinik,  "On the Cauchy problem for weakly hyperbolic equations"  ''Comm. Pure Appl. Math.'' , '''23'''  (1970)  pp. 569–586</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  pp. Chapt. XII {{MR|0717035}} {{MR|0705278}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''III''' , Springer  (1985)  pp. Chapt. XXIII {{MR|1540773}} {{MR|0781537}} {{MR|0781536}} {{ZBL|0612.35001}} {{ZBL|0601.35001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  I.G. Petrovskii,  "Ueber das Cauchysche Problem für Systeme von partiellen Differentialgleichungen"  ''Mat. Sb. (N.S.)'' , '''2(44)'''  (1937)  pp. 815–870 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Mizohata,  "The theory of partial differential equations" , Cambridge Univ. Press  (1973)  (Translated from Japanese) {{MR|0599580}} {{ZBL|0263.35001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Chaillou,  "Hyperbolic differential polynomials" , Reidel  (1979) {{MR|0557901}} {{ZBL|0424.35055}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Chazarain,  "Opérateurs hyperboliques à characteristique de multiplicité constante"  ''Ann. Inst. Fourier'' , '''24'''  (1974)  pp. 173–202 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Gårding,  "Linear hyperbolic equations with constant coefficients"  ''Acta Math.'' , '''85'''  (1951)  pp. 1–62 {{MR|41336}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  O.A. Oleinik,  "On the Cauchy problem for weakly hyperbolic equations"  ''Comm. Pure Appl. Math.'' , '''23'''  (1970)  pp. 569–586 {{MR|0264227}} {{ZBL|}} </TD></TR></table>

Latest revision as of 22:11, 5 June 2020


at a given point $ M( x _ {1} \dots x _ {n} ) $

A partial differential equation for which the Cauchy problem is uniquely solvable for initial data specified in a neighbourhood of $ M $ on any non-characteristic surface (cf. Characteristic surface). In particular, a partial differential equation for which the normal cone has no imaginary zones is a hyperbolic partial differential equation. The differential equation

$$ \tag{* } L ( u) = H ( D _ {1} \dots D _ {n} ) u + F ( D _ {1} \dots D _ {n} ) u + G ( x) = 0, $$

where $ D _ {i} = \partial / \partial x _ {i} $( $ i = 1 \dots n $), $ H( D _ {1} \dots D _ {n} ) $ is a homogeneous polynomial of degree $ m $, while the polynomial $ F $ is of lower degree than $ m $, is a hyperbolic partial differential equation if its characteristic equation

$$ Q ( \xi _ {1} \dots \xi _ {n} ) = H ( \xi _ {1} \dots \xi _ {n} ) = 0 $$

has $ m $ different real solutions with respect to one of the variables $ \xi _ {1} \dots \xi _ {n} $, the remaining ones being fixed. Any equation (*) of the first order $ ( m = 1 ) $ with real coefficients is a hyperbolic partial differential equation. A second-order equation

$$ L ( u) = u _ {tt} - \sum _ {i, j = 1 } ^ { n } a _ {ij} D _ {i} D _ {j} u + Fu + G = 0 $$

is hyperbolic if the quadratic form

$$ \sum _ {i, j = 1 } ^ { n } a _ {ij} \xi _ {i} \xi _ {j} $$

is positive definite.

Comments

The special variable among the $ \xi _ {1} \dots \xi _ {n} $ such that $ H ( \xi _ {1} \dots \xi _ {n} ) $ has $ m $ different real solutions for each set of fixed values of the other $ n - 1 $ is often taken to be $ t $( time). One speaks then of a (strictly) hyperbolic equation or an equation of (strictly) hyperbolic type with respect to the $ t $- direction. More generally one considers hyperbolicity with respect to a vector $ N $[a1].

A polynomial $ P $ of degree $ m $ with principal part $ P _ {m} $ is called hyperbolic with respect to the real vector $ N $ if $ P _ {m} ( N) \neq 0 $ and there exists a number $ \tau _ {0} > 0 $ such that

$$ P ( \xi + i \tau N) \neq 0 \ \ \textrm{ if } \ \xi \in \mathbf R ^ {n} ,\ \tau < \tau _ {0} . $$

If $ P _ {m} $ is such that $ P _ {m} ( N) \neq 0 $ and $ P _ {m} ( \xi + \tau N) $ has only simple real roots for every real $ \xi \neq 0 $, then $ P $ is said to be strictly hyperbolic or hyperbolic in the sense of Petrovskii.

The Cauchy problem for a constant-coefficient differential operator $ P $ with data on a non-characteristic plane is well posed for arbitrary lower-order terms if and only if $ P $ is strictly hyperbolic. For a discussion of similar matters for polynomials $ P $ with variable coefficients cf. [a2].

For a system of higher-order linear partial differential equations

$$ \sum _ {j = 1 } ^ { l } \sum _ {| \alpha | \leq N _ {j} } a _ \alpha ^ {ij} ( x) \frac{\partial ^ \alpha }{\partial x ^ \alpha } u _ {j} = 0,\ \ i = 1 \dots l , $$

where $ \alpha = ( \alpha _ {0} \dots \alpha _ {n} ) $, is a hyperbolic system of partial differential equations in the sense of Petrovskii if the determinant

$$ \mathop{\rm det} \ \left ( \sum _ {| \alpha | \leq N _ {j} } a _ \alpha ^ {ij} \frac{\partial ^ \alpha }{\partial x ^ \alpha } \right ) $$

calculated in the ring of differential operators is a hyperbolic polynomial in the sense of Petrovskii (as a polynomial of degree $ N = \sum N _ {j} $). The Cauchy problem for a system that is hyperbolic in this sense is well posed [a3], [a4].

Instead of strictly hyperbolic one also finds the term strongly hyperbolic and instead of hyperbolic also weakly hyperbolic (which is therefore the case in which the lower-order terms of $ P $ do matter).

References

[a1] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) pp. Chapt. XII MR0717035 MR0705278 Zbl 0521.35002 Zbl 0521.35001
[a2] L.V. Hörmander, "The analysis of linear partial differential operators" , III , Springer (1985) pp. Chapt. XXIII MR1540773 MR0781537 MR0781536 Zbl 0612.35001 Zbl 0601.35001
[a3] I.G. Petrovskii, "Ueber das Cauchysche Problem für Systeme von partiellen Differentialgleichungen" Mat. Sb. (N.S.) , 2(44) (1937) pp. 815–870
[a4] S. Mizohata, "The theory of partial differential equations" , Cambridge Univ. Press (1973) (Translated from Japanese) MR0599580 Zbl 0263.35001
[a5] J. Chaillou, "Hyperbolic differential polynomials" , Reidel (1979) MR0557901 Zbl 0424.35055
[a6] J. Chazarain, "Opérateurs hyperboliques à characteristique de multiplicité constante" Ann. Inst. Fourier , 24 (1974) pp. 173–202
[a7] L. Gårding, "Linear hyperbolic equations with constant coefficients" Acta Math. , 85 (1951) pp. 1–62 MR41336
[a8] O.A. Oleinik, "On the Cauchy problem for weakly hyperbolic equations" Comm. Pure Appl. Math. , 23 (1970) pp. 569–586 MR0264227
How to Cite This Entry:
Hyperbolic partial differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_partial_differential_equation&oldid=16785
This article was adapted from an original article by B.L. Rozhdestvenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article