Linear hyperbolic partial differential equation and system

A partial differential equation (or system) of the form

$$ \tag{1 } \sum _ {| \alpha | \leq m } a _ \alpha ( x) D ^ \alpha u = f $$

for which at any point $ x = ( x _ {0}, \dots, x _ {n} ) $ of its domain of definition $ \Omega $ one can distinguish among the real variables $ y _ {0}, \dots, y _ {n} $ (if necessary, after a suitable affine transformation of the independent variables) one variable, say $ y _ {0} = \lambda $, in a such a way that for all points $ Y = ( y _ {1}, \dots, y _ {n} ) $ of the $ n $-dimensional Euclidean space $ \mathbf R ^ {n} $ the characteristic equation (in $ \lambda $)

$$ \tag{2 } \mathop{\rm det} \left . \sum _ {| \alpha | \leq m } a _ \alpha ( x) y ^ \alpha \right | _ {y _ {0} = \lambda } ,\ \ y ^ \alpha = ( y _ {0} ^ {\alpha _ {0} }, \dots, y _ {n} ^ {\alpha _ {n} } ) , $$

has exactly $ N m $ real roots. Here $ \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} = \partial / \partial x _ {j} $, $ j = 0, \dots, n $, $ m $ is the order of the system (1), $ a _ \alpha ( x) $ is a real square matrix of order $ N $, defined in $ \Omega $, $ u ( x) = \| u _ {j} ( x) \| $, $ j = 1, \dots, N $, is an unknown column vector, and $ f ( x) $ is a vector with $ N $ components, defined in $ \Omega $.

A typical example is the wave equation

$$ \tag{3 } u _ {x _ {0} x _ {0} } = \ \sum _ { j= 1} ^ { n } u _ {x _ {j} x _ {j} } . $$

Many problems in mathematical physics reduce to linear hyperbolic partial differential equations or systems of equations.

A subset $ S : \phi ( x) = 0 $ is said to be characteristic at a point $ x $ if $ \mathop{\rm grad} \phi \neq 0 $ and $ Q ( x , \mathop{\rm grad} \phi ) = 0 $, where

$$ Q ( x , y ) \equiv Q ( x , y _ {0} , Y ) = \mathop{\rm det} \ \sum _ {| \alpha | = m } a _ \alpha ( x) y ^ \alpha $$

is the characteristic form of the system (1). If $ Q ( x , y ) = 0 $, then one says that the vector $ y $ defines a characteristic direction or characteristic normal at the point $ x $. The surface $ S $ is called a characteristic surface (or characteristic) of the system (1) if

$$ Q ( x , \mathop{\rm grad} \phi ) = 0 \ \ \textrm{ for all } x \in S . $$

A surface which does not have characteristic normals at any point is called a free surface. On a free surface the rank of the characteristic matrix

$$ A ( x , y ) \equiv A ( x , y _ {0} , Y ) = \ \sum _ {| \alpha | = m } a _ \alpha ( x) y ^ \alpha ,\ \ y = \mathop{\rm grad} \phi , $$

is equal to $ N $, while on a characteristic surface $ S $ it is less than $ N $. A characteristic $ S $ is said to be simple if for some $ j $ and all $ x \in S $,

$$ \left ( \frac \partial {\partial x _ {j} } \right ) A ( x , \mathop{\rm grad} \phi ) \neq 0 . $$

Otherwise the characteristic is said to be multiple. A characteristic is sometimes said to be simple if the rank of the matrix $ A ( x , \mathop{\rm grad} \phi ) $ is $ N - 1 $.

The system (1) is said to be hyperbolic at the point $ x $ with respect to the hyperplane $ S _ {0} $: $ x _ {0} = 0 $ if the matrix $ A ( x , 1 , Y ) = a _ {m , 0, \dots, 0 } $ is non-singular (that is, the surface $ S $ is free) and if all roots $ \lambda = \lambda _ {k} $, $ k = 1, \dots, m N $, of the characteristic equation $ Q ( x , \lambda , Y ) = 0 $ are real for all points $ Y \in \mathbf R ^ {n} $. The system (1) is said to be hyperbolic in the domain $ \Omega $ with respect to $ S _ {0} $ if it is hyperbolic with respect to $ S _ {0} $ at every point $ x \in \Omega $.

An important class of hyperbolic equations and systems consists of strictly hyperbolic equations and systems, which are sometimes called fully hyperbolic systems, or systems, hyperbolic in the narrow sense. A system (1) is called a strictly hyperbolic system if all roots $ \lambda = \lambda _ {k} $ of the characteristic equation are distinct for any non-zero vector $ Y \in \mathbf R ^ {n} $. The characteristics of a strictly hyperbolic equation (or system) are simple. Strictly hyperbolic (with respect to $ S _ {0} $) systems are notable for the fact that the Cauchy problem

$$ \tag{4 } D _ {0} ^ {k} u \mid _ {x _ {0} = 0 } = \ \psi _ {k} ( x) ,\ k = 0, \dots, m - 1 , $$

for them is well-posed under the single assumption of sufficient smoothness of the coefficients $ a _ \alpha ( x) $, $ f ( x) $ of the system (1) and of the initial data (initial functions) $ \psi _ {k} ( x) $, $ x = ( x _ {0}, \dots, x _ {n} ) $. There are examples of hyperbolic but not strictly hyperbolic equations of the form (1) (even with constant coefficients in front of the derivatives of order $ m $) for which the Cauchy problem is ill-posed.

The solution $ u ( x) $ of the Cauchy problem (4) for the wave equation (3) can be written out explicitly, and when $ m \equiv 0 ( \mathop{\rm mod} 2) $, and only then, it has the property that the value of $ u ( x) $ at the vertex $ x ^ {*} $ of the characteristic cone $ | X - X ^ {*} | = x _ {0} ^ {*} - x _ {0} $, $ X = ( x _ {1}, \dots, x _ {n} ) $, depends only on the value of the initial data $ \psi _ {0} ( x) $ and $ \psi _ {1} ( x) $ and their derivatives on the base $ | X - X ^ {*} | = x _ {0} ^ {*} $, $ x _ {0} = 0 $, of this cone (the so-called Huygens principle).

For strictly hyperbolic (with respect to $ S _ {0} $) equations and systems the question of the diffusion of waves and the related question of gaps have been investigated (cf. Lacuna). An exhaustive answer has been given in the case of an equation with constant coefficients of the form

$$ \sum _ {| \alpha | = m } a _ \alpha D ^ \alpha u = 0 ,\ \ a _ \alpha = \textrm{ const } . $$

For the case of one (scalar) equation with constant coefficients $ a _ \alpha ( x) = a _ \alpha = \textrm{ const } $ the definition of strict hyperbolicity has been generalized as follows. Equation (1) is said to be hyperbolic with respect to a non-zero vector $ y \in \mathbf R ^ {n+ 1} $ if

$$ \sum _ {| \alpha | = m } a _ \alpha y ^ \alpha \neq 0 $$

and if there is a real number $ \lambda _ {0} $ such that for all $ \xi = ( \xi _ {0}, \dots, \xi _ {n} ) \in \mathbf R ^ {n+} 1 $ and $ \lambda > \lambda _ {0} $,

$$ \sum _ {| \alpha | \leq m } a _ \alpha ( \lambda y + i \xi ) ^ \alpha \neq 0 . $$

Of all linear equations with constant coefficients, only for equations that are hyperbolic in this sense the Cauchy problem is well-posed for arbitrary sufficiently smooth initial functions defined on the hyperplane

$$ \sum _ { j= 0} ^ { n } x _ {j} y _ {j} = 0 . $$

In particular, the wave equation (3) is hyperbolic in this sense with respect to any vector for which

$$ y _ {0} ^ {2} > \sum _ { j= 1} ^ { n } y _ {j} ^ {2} . $$

There are various generalizations of the definition of strict hyperbolicity of equations and systems. These are mainly equations and systems that are completely characterized by the fact that the Cauchy problem with data on a free surface is uniquely solvable for them for any sufficiently smooth initial functions, without any restrictions on the growth at infinity.

Another important class of linear hyperbolic systems of the first order is the class of symmetric hyperbolic systems. The system

$$ \tag{5 } \sum _ { j= 0} ^ { n } a _ {j} ( x) u _ {x _ {j} } + b ( x) u = f ( x) , $$

where $ a _ {j} ( x) $, $ b ( x) $ are square matrices of order $ N $, defined in $ \Omega $, and $ u ( x) $ is an unknown vector of $ N $ components, is called a symmetric hyperbolic system in $ \Omega $ if the matrices $ a _ {j} ( x) $ are symmetric (or are symmetrizable simultaneously by the same transformation) and if at every point there is a spatially-oriented hyperplane (or, space-like hyperplane), that is, a hyperplane whose normal $ y = ( y _ {0}, \dots, y _ {n} ) $ has the property that the matrix $ \sum _ {j= 0} ^ {n} y _ {j} a _ {j} ( x) $ is positive definite. If for a symmetric hyperbolic system (5) with sufficiently smooth coefficients the given initial functions and the right-hand side have square-integrable generalized partial derivatives of order $ p $, then there is a unique generalized solution of the Cauchy problem with the same number of square-integrable partial derivatives. Any strictly hyperbolic partial differential equation of the second order reduces to a symmetric hyperbolic system.

An equation (1) of the second order in the class of solutions regular in a domain $ \Omega $ can be written in the form

$$ \tag{6 } \sum _ {k , j = 0 } ^ { n } a _ {kj} ( x) u _ {x _ {k} x _ {j} } + \sum _ { j= 0} ^ { n } b _ {j} ( x) u _ {x _ {j} } + c ( x) u = f ( x) , $$

where $ a _ {kj} ( x) = a _ {jk} ( x) $, $ b _ {j} ( x) $, $ c ( x) $, and $ f ( x) $ are functions defined in $ \Omega $. Equation (6) is hyperbolic in $ \Omega $ if at every point of $ \Omega $ all eigen values of the matrix of leading coefficients $ a ( x) = \| a _ {jk} ( x) \| $, $ k , j = 0, \dots, n $, are non-zero, and one of these eigen values differs in sign from all others. With respect to (6), along with the characteristic surface one can distinguish two types of smooth surfaces: spatially-oriented surfaces and time-oriented surfaces (also called space-like and time-like surfaces). If the surfaces are given by an equation of the form $ \phi ( x) = 0 $, then on a surface of the first type $ Q ( x , \mathop{\rm grad} \phi ) > 0 $, while on a surface of the second type $ Q ( x , \mathop{\rm grad} \phi ) < 0 $, where

$$ Q ( x , y ) = \sum _ {k , j = 0 } ^ { n } a _ {kj} ( x) y _ {k} y _ {j} . $$

The Cauchy problem for hyperbolic partial differential equations with initial data on a time-like surface is generally not well-posed.

A hyperbolic partial differential equation

$$ u _ {x _ {0} x _ {0} } - \sum _ {k , j = 1 } ^ { n } a _ {kj} ( x) u _ {x _ {k} x _ {j} } + \sum _ { j= 0} ^ { n } b _ {j} ( x) u _ {x _ {j} } + c ( x) u = f ( x) $$

is said to be uniformly (or regularly) hyperbolic in a domain $ \Omega $ if there is a positive number $ \epsilon $ such that

$$ \sum _ {k , j = 1 } ^ { n } a _ {kj} ( x) \xi _ {k} \xi _ {j} > \ \epsilon \sum _ { j= 1} ^ { n } \xi _ {j} ^ {2} $$

for all $ x $ in $ \overline \Omega \; $ and for any non-zero vector $ ( \xi _ {1}, \dots, \xi _ {n} ) \in \mathbf R ^ {n} $. For $ n = 1 $ the inequality

$$ a _ {01} ^ {2} - a _ {00} a _ {11} < 0 \ \ \textrm{ for all } x \in \overline \Omega $$

is a necessary and sufficient condition for (6) to be uniformly hyperbolic in $ \Omega $. The equation for the vibration of a string,

$$ u _ {x _ {0} x _ {0} } - u _ {x _ {1} x _ {1} } = 0, $$

is a typical representative of a linear uniformly hyperbolic partial differential equation of the second order with two independent variables. The general solution of this equation in any convex domain $ \Omega $ of the plane $ \mathbf R ^ {2} $ is given by the d'Alembert formula:

$$ u ( x) = f ( x _ {0} + x _ {1} ) + g ( x _ {0} - x _ {1} ) , $$

where $ f $ and $ g $ are arbitrary functions.

After a non-singular real change of variables $ x _ {0} $ and $ x _ {1} $, the hyperbolic partial differential equation (6) with $ n = 1 $ reduces to the normal (canonical) form

$$ \tag{7 } u _ {y _ {0} y _ {0} } - u _ {y _ {1} y _ {1} } + A ( y) u _ {y _ {0} } + B ( y) u _ {y _ {1} } + C ( y) u = F ( y) , $$

$$ y = ( y _ {0} , y _ {1} ) . $$

For hyperbolic systems written in the form (7), where $ A ( y) $, $ B ( y) $ and $ C ( y) $ are given real square matrices of order $ N $, $ F ( y) $ is a given vector and $ u $ an unknown vector, both with $ N $ components, the question of the well-posedness of the Cauchy problem with initial data on a non-characteristic (free) curve and the Goursat problem with data on two intersecting characteristics have been completely investigated by the Riemann method.

The main problems about hyperbolic equations are the following: the Cauchy problem, the Cauchy characteristic problem and the mixed problem (see also Mixed and boundary value problems for hyperbolic equations and systems).

In the investigation of the main problems an important role is played by fundamental solutions, which make it possible to obtain explicit (integral) representations of regular and generalized solutions and to establish their structural and qualitative properties, in particular to study the question of wave fronts and the propagation of discontinuities.

Equation (6) is called an ultra-hyperbolic equation in a domain $ \Omega $ if at every point $ x \in \Omega $ all eigenvalues of the matrix $ a ( x) $ are non-zero and at least two of them differ in sign from all the others, of which there are at least two. An example of an ultra-hyperbolic equation is an equation of the form

$$ \tag{8 } \sum _ { j= 0} ^ { n } ( u _ {x _ {j} x _ {j} } - u _ {y _ {j} y _ {j} } ) = 0 ,\ \ n \geq 1 , $$

which has the following property: If $ u ( x , y ) $ is a regular solution of (8) in a domain $ \Omega $ of the Euclidean space of the variables $ x = ( x _ {0}, \dots, x _ {n} ) $, $ y = ( y _ {0}, \dots, y _ {n} ) $ and if $ ( x ^ {*} , y ^ {*} ) $ is an arbitrary point of $ \Omega $, then the mean value of the function $ u ( x , y ^ {*} ) $ calculated on the sphere $ \sum _ {j= 0} ^ {n} ( x _ {j} - x _ {j} ^ {*} ) ^ {2} = r ^ {2} $ with centre at the point $ x ^ {*} = ( x _ {0} ^ {*}, \dots, x _ {n} ^ {*} ) $ and radius $ r $, is equal to the mean value of the function $ u ( x ^ {*} , y ) $ calculated on the sphere $ \sum _ {j= 0} ^ {n} ( y _ {j} - y _ {j} ^ {*} ) ^ {2} = r ^ {2} $ with centre at the point $ y ^ {*} = ( y _ {0} ^ {*}, \dots, y _ {n} ^ {*} ) $ and the same radius $ r $. This theorem is extensively used in the theory of linear hyperbolic partial differential equations of the second order with constant coefficients.

References

[1]	J. Hadamard, "Lectures on Cauchy's problem in linear partial differential equations" , Dover, reprint (1952)
[2]	L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964) MR0163043 Zbl 0126.00207
[3]	A.V. Bitsadse, "Equations of mixed type" , Pergamon (1964) (Translated from Russian)
[4]	A.V. Bitsadze, "Equations of mathematical physics" , MIR (1980) (Translated from Russian) MR0587310 MR0581247 Zbl 0499.35002
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[6]	I.M. Gel'fand, G.E. Shilov, "Generalized functions" , Acad. Press (1964) (Translated from Russian) MR435831 Zbl 0115.33101
[7]	L. Gårding, "Cauchy's problem for hyperbolic equations" , Third Congress Scand. Math. Helsinki, 1957 , Mercators Tryckeri (1958) pp. 104–109 MR116143
[8]	A.A. Dezin, "Boundary problems for certain symmetric linear first-order systems" Mat. Sb. , 49 : 4 (1959) pp. 459–484 (In Russian)
[9]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[10]	O.A. Ladyzhenskaya, "A mixed problem for a hyperbolic equation" , Moscow (1953) (In Russian)
[11]	J. Leray, L. Gårding, T. Kotake, "Uniformisation et developpement asymptotique de la solution du problème de Cauchy linéaire à données holomorphes; analogie avec la théorie des ondes asymptotiques et approchées" Bull. Soc. Math. France , 92 (1964) pp. 236–361 MR0196280 Zbl 0156.33101 Zbl 0147.08101
[12]	I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian) MR0211021
[13]	V.I. Smirnov, "A course of higher mathematics" , 2 , Addison-Wesley (1964) (Translated from Russian) MR0182690 MR0182688 MR0182687 MR0177069 MR0168707 Zbl 0122.29703 Zbl 0121.25904 Zbl 0118.28402 Zbl 0117.03404
[14]	A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) MR104888
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Comments

A fully hyperbolic system, or system hyperbolic in the narrow sense, is also called a totally hyperbolic system or a system hyperbolic in the sense of Petrovskii.

For the definition of hyperbolicity in the case of scalar equations with constant coefficients, see [a1], Vol. II, Chapt. 12.

For fundamental solutions and propagation of singularities, see also [a1], Vol. III, Chapts. 13, 14.

The theorem mentioned in the last line of the main article above is called Asgeirsson's theorem, cf. [9], Chapt. 6.

References

[a1]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1–4 , Springer (1983–1985) MR2512677 MR2304165 MR2108588 MR1996773 MR1481433 MR1313500 MR1065993 MR1065136 MR0961959 MR0925821 MR0881605 MR0862624 MR1540773 MR0781537 MR0781536 MR0717035 MR0705278 Zbl 1178.35003 Zbl 1115.35005 Zbl 1062.35004 Zbl 1028.35001 Zbl 0712.35001 Zbl 0687.35002 Zbl 0619.35002 Zbl 0619.35001 Zbl 0612.35001 Zbl 0601.35001 Zbl 0521.35002 Zbl 0521.35001
[a2]	P.R. Garabedian, "Partial differential equations" , Wiley (1964) MR0162045 Zbl 0124.30501
[a3]	F. John, "Partial differential equations" , Springer (1974) MR2098409 MR1713869 MR1185075 MR0765229 MR0831655 MR0614918 MR0598466 MR0514404 MR0502136 MR0470380 MR0447820 MR0390518 MR0350162 MR1536882 MR0304828 MR0261133 MR1534312 MR0208121 MR0163043 MR0130456 Zbl 0304.90142