Difference between revisions of "Hyperbolic partial differential equation"
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+ | $#C+1 = 42 : ~/encyclopedia/old_files/data/H048/H.0408300 Hyperbolic partial differential equation | ||
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− | + | ''at a given point $ M( x _ {1} \dots x _ {n} ) $'' | |
− | + | 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 | ||
− | + | $$ \tag{* } | |
+ | L ( u) = H ( D _ {1} \dots D _ {n} ) u + | ||
+ | F ( D _ {1} \dots D _ {n} ) u + G ( x) = 0, | ||
+ | $$ | ||
− | + | 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 | is hyperbolic if the quadratic form | ||
− | + | $$ | |
+ | \sum _ {i, j = 1 } ^ { n } a _ {ij} \xi _ {i} \xi _ {j} $$ | ||
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. | ||
− | + | 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. [[#References|[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 | + | 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 | + | 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 |
Hyperbolic partial differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperbolic_partial_differential_equation&oldid=16785