Quasi-linear hyperbolic equations and systems

Differential equations and systems of differential equations of the form

$$ \tag{1 } L [ u ] \equiv \ \sum _ {| \alpha | \leq m } a ^ \alpha D ^ \alpha u = f . $$

The system of equations (1) is to be solved for the vector-valued function $ u ( x) $ with components $ u _ {1} ( x), \dots, u _ {k} ( x) $ (in the case of a single equation, $ k = 1 $). The coefficients $ a ^ \alpha $ are matrices whose entries depend on the independent space variables $ x = ( x _ {0}, \dots, x _ {n} ) $, the vector-valued function $ u $ and its partial derivatives up to order $ m - 1 $ inclusive. The right-hand side $ f $ also depends on these arguments. If the $ a ^ \alpha $ are square matrices of the same order as the number of components of $ u $, the system (1) is called a determined system. The characteristic form

$$ Q ( \xi ) = Q ( \xi _ {0}, \dots, \xi _ {n} ) = \mathop{\rm det} \ \left \| \sum _ {| \alpha | = m } a ^ \alpha \xi ^ \alpha \ \right \| $$

is determined by the principal part $ \sum _ {| \alpha | = m } a ^ \alpha D ^ \alpha $ of $ L $. Here $ D ^ \alpha = \partial ^ {| \alpha | } / \partial x _ {0} ^ {\alpha _ {0} } \cdots \partial x _ {n} ^ {\alpha _ {n} } $ and $ \xi ^ \alpha = \xi _ {0} ^ {\alpha _ {0} } \cdots \xi _ {n} ^ {\alpha _ {n} } $.

The hyperbolicity of the system (1) is defined by the following characterization of the operator $ L $. At each set of values for $ x $, $ u $ and its derivatives up to order $ m- 1 $ inclusive there exists a vector $ \zeta \in \mathbf R ^ {n+1} $ such that for any $ \eta \in \mathbf R ^ {n+1} $ that is not parallel to $ \zeta $, the characteristic equation

$$ \tag{2 } Q ( \lambda \zeta + \eta ) = 0 $$

has $ mk $ real roots $ \lambda $ (each root counted as many times as its multiplicity).

The surface elements passing through some point $ P \in \mathbf R ^ {n+1} $ and orthogonal to a vector $ \zeta $ are called space like and the direction of the normal to a space-like surface is called time like.

A curve that has at each point $ x $ a time-like tangent is called a time-like curve.

Among the problems of quasi-linear hyperbolic equations and systems a central place is occupied by the Cauchy problem, which is the problem of finding the solution $ u $ of the system (1) when on some smooth $ n $-dimensional hypersurface $ \Pi $ defined by an equation

$$ \phi ( x) = 0 ,\ \ | D \phi | = | \mathop{\rm grad} \phi | \neq 0 , $$

the values are known of the function $ u $ and its partial derivatives up to order $ m - 1 $ (inclusive) along some direction not tangent to $ \Pi $. If such a solution can always be found, then $ \Pi $ is said to be a free hypersurface with respect to $ L $.

If the coefficients (1) and the Cauchy conditions given on an analytic free hypersurface $ \Pi $ are analytic, then an analytic solution in a neighbourhood of $ \Pi $ is unique; if in addition the Cauchy conditions involve all derivatives up to order $ m - 1 $ on $ \Pi $, then, according to the Cauchy–Kovalevskaya theorem, an analytic solution exists in a sufficiently small neighbourhood of any point of $ \Pi $. This theorem is applicable to all analytic equations and systems, irrespective of their type. For non-analytic data the theorem does not given an answer to the problem of existence of such solutions.

If no part of $ \Pi $ is free, $ \Pi $ is called a characteristic surface with respect to $ L $ (cf. also Characteristic). In order that a hypersurface $ \Pi $ be characteristic with respect to $ L $ it is necessary and sufficient that the equation

$$ \tag{3 } Q ( \xi ) = 0 $$

holds, where $ \xi = ( \xi _ {0}, \dots, \xi _ {n} ) $ is a normal to $ \Pi $. The characteristic matrix, the characteristic form and the characteristic surface are invariant with respect to non-singular transformations. Characteristic surfaces play an important role in the study of many problems for equations and systems of type (1).

Many problems of mathematical physics reduce to symmetric hyperbolic equations and systems. A first-order quasi-linear system

$$ \tag{1'} L [ u ] \equiv \ \sum _ { i= 0} ^ { n } a ^ {i} u _ {x _ {i} } + b u = f $$

is called symmetric if all matrices $ a ^ {i} $ are symmetric. A symmetric system (1'}) is called a symmetric hyperbolic system at a point $ P $ if one of the matrices $ a ^ {i} $ or some linear combination of the $ \xi ^ {i} a ^ {i} $ is of fixed sign. A single equation or system of equations of higher order is called symmetric hyperbolic if it is equivalent to some symmetric hyperbolic system of order one. A system (1'}) with a positive-definite coefficient $ a ^ {0} $ can be reduced by a linear transformation to the form

$$ \tag{1\prime\prime } u _ {t} + M [ u ] = f , $$

where $ x _ {0} $ is replaced by $ t $ and $ M $ is a symmetric operator. Using energy inequalities (cf. Energy inequality) and an iteration method, the existence can be proved of a solution to the Cauchy problem for quasi-linear second-order systems for a single non-linear equation of arbitrary order.

Since the characteristics and bicharacteristics (cf. Bicharacteristic) for hyperbolic equations and systems are defined for linear and quasi-linear ones in the same way (cf. also Linear hyperbolic partial differential equation and system), the well-known facts about the distribution of discontinuities of the lower-order derivatives of the solutions of linear equations and systems hold also for the case of quasi-linear equations. The distribution of discontinuities of the solution $ u $ of a hyperbolic system have been considered for conservation laws, that is, for systems of order one of the following form:

$$ \tag{4 } \sum _ { i= 0} ^ { n } \frac \partial {\partial x _ {i} } F ^ { ij } ( x , u ) = 0 ,\ \ j = 1, \dots, k , $$

under the condition that the surface of the shock wave is non-characteristic (cf. also Hyperbolic partial differential equation).

Many papers are devoted to the study of quasi-hyperbolic equations and systems in the case of two spatial variables $ x $ and $ t $. An important part of these studies concerns quasi-linear systems of order one of the form:

$$ \tag{5 } L [ u ] \equiv A u _ {x} + B u _ {t} = C , $$

where $ A $ and $ B $, which are square matrices of order $ k $, and the vector $ C $ depend on $ x $, $ t $ and on the vector $ u ( x , t ) = \{ u _ {1} ( x , t ), \dots, u _ {k} ( x , t ) \} $. If it is only the right-hand side $ C $ that depends on the unknown function $ u ( x , t ) $, then (5) is called an almost-linear system. Almost-linear first-order systems can be reduced by non-singular linear transformations to symmetric systems. Under the hypothesis that $ \mathop{\rm det} B \neq 0 $, the quasi-linear system (5) has the following form:

$$ \tag{6 } u _ {t} + A u _ {x} = C . $$

The matrix $ A $ can be transformed to diagonal form with entries $ \kappa _ {i} $, where the vectors $ ( \kappa _ {i} , 1 ) $ define the characteristic directions; this way of writing is called the normal form of the system (5). The uniqueness theorem holds for systems of the form (6), independently of whether or not their characteristics depend on the solution $ u ( x , t ) $.

If the coefficients of $ A $, $ C $ and the initial value of the solution $ u ( x , t ) $ at $ t = 0 $ have first-order derivatives in all arguments and if these derivatives satisfy a Lipschitz condition, then one can find a neighbourhood $ 0 < t < h $ of the interval of the axis $ t = 0 $ in which there exists a unique solution of the Cauchy problem with first-order derivatives satisfying a Lipschitz condition. Here one introduces vector-valued functions $ v ( x , t ) $ that coincide at $ t = 0 $ with the given initial data and such that their first-order derivatives satisfy a Lipschitz condition. After substituting $ v ( x , t ) $ into the coefficients of $ A $ and $ C $, the Cauchy problem is solved with the given initial data for the linear equation. Corresponding to each function $ v ( x , t ) $ there exists a solution $ u ( x , t ) $, and the solution of the problem for the system (6) is found as the fixed element of the transformation $ u = \mathfrak A [ v ] $, or as the limit of the uniformly-convergent sequence $ \{ u _ {n} \} _ {n=1} ^ \infty $ obtained from the iterative process

$$ u _ {n+1 } \equiv \mathfrak A [ u _ {n} ] . $$

A similar result is obtained in case the coefficients of $ A $, the vector $ C $ and the initial data are sufficiently smooth. Reducing the problem to a system of integral equations, solvability is proved on the basis of the contracting-mapping principle (contraction-mapping principle).

The first-order quasi-linear system (5) is called a weakly non-linear system if the corresponding normal form

$$ \tag{7 } u _ {t} + A u _ {x} = C $$

is such that every entry $ a ^ {ii} $ of the diagonal matrix satisfies the condition:

$$ \frac \partial {\partial u _ {i} } a ^ {ii} ( x , t , u _ {1}, \dots, u _ {k} ) = 0 ,\ \ i = 1, \dots, k . $$

Otherwise (5) is called a strongly non-linear system. If the solution of a weakly non-linear system for $ k = 2 $ is bounded, then the Cauchy problem is always solvable in the domain of definition.

Apart from the initial value problem, other problems have been posed and investigated for first-order quasi-linear systems. For example, so-called mixed problems (cf. Mixed and boundary value problems for hyperbolic equations and systems), consisting in the determination of a solution $ u ( x , t ) $ of the system (7) which, along with initial conditions $ u ( x , 0 ) = \psi ( x) $ for $ t = 0 $, $ \alpha \leq x \leq \beta $, satisfies also boundary conditions on certain smooth curves $ \Gamma _ {1} $ and $ \Gamma _ {2} $ emanating from the points $ ( \alpha , 0 ) $ and $ ( \beta , 0 ) $, respectively. These boundary conditions can be either linear or non-linear with respect to the solution $ u ( x , t ) $. In certain cases, when special conditions are observed, the well-posedness of problems of the above type has been successfully verified.

For a strongly non-linear system (6), the following problem has been studied for the case $ k = 2 $: Let $ \Gamma _ {1} $ be a curve emanating from the coordinate origin and given by the equation $ x = x ( t) $, and suppose that $ \Gamma _ {1} $ is characteristic and has a continuous tangent. It is required to determine the solution of (6) in some domain bounded by $ \Gamma _ {1} $ and another curve $ \Gamma _ {2} $ also passing through the origin, while the solution $ u ( x , t ) $ is known on $ \Gamma _ {1} $, and under the extra condition that one of the components of the vector $ u ( x , t ) $ has a specific singularity at the point of intersection of the boundary curves. The solvability of similar problems for $ t \leq \delta $, for sufficiently small $ \delta $, is known.

Among the problems posed for quasi-linear first-order hyperbolic systems there are a number of important applied problems: the problem of decomposing an arbitrary discontinuity, problems from chromatography and filtration, etc.

The results set out above carry a local character, and relate only to regular solutions. If, in fact, the solutions are not differentiable or are even discontinuous functions, then one introduces so-called weak solutions by various methods; for these, global uniqueness and existence theorems are proved for the solution of the Cauchy problem. As a rule these solutions are defined in fairly broad classes of functions (bounded measurable functions, locally summable functions, etc.). They satisfy the equation or system in a certain sense, for example in the sense of the theory of generalized functions, or are subject to completely determinate integral relations. In certain cases a weak solution is called a solution of the Cauchy problem if the difference between the solution and the initial data converges weakly to zero as the point $ ( t, x) $ converges to the support of the initial data.

Some methods of constructing global weak solutions of the Cauchy problem are known for conservation laws with initial data of a fairly general type. For example, smoothing methods and finite-difference schemes for solving initial value problems.

General non-linear hyperbolic equations and systems can be reduced to quasi-linear first-order hyperbolic systems by differentiating with respect to the independent variables. Any second-order hyperbolic equation can be reduced to a first-order symmetric hyperbolic system, and the facts relating to first-order hyperbolic systems remain valid also for a single second-order hyperbolic equation.

An existence theorem for the solution of the Cauchy problem for a single hyperbolic equation of higher order was obtained under the requirement of a fairly high smoothness of the coefficients of the equation (see [9]).

The Cauchy problem for hyperbolic quasi-linear equations of higher order has been studied by reducing it to a similar problem for quasi-linear systems of order one. For second-order equations one additionally uses a second method, consisting of the introduction of a characteristic coordinate system $ ( \alpha , \beta ) $. The variables $ x , t $, the function $ u $ and its derivatives of order one and two are regarded as functions of the characteristic variables, and a system of equations is obtained. This system consists of six first-order equations in the functions

$$ x = x ( \alpha , \beta ) ,\ \ y = y ( \alpha , \beta ) ,\ \ u = u ( \alpha , \beta ) , $$

$$ p = u _ {x} ( \alpha , \beta ) ,\ q = u _ {y} ( \alpha , \beta ) , $$

one of which follows from all the others. It is possible to consider this system of five quasi-linear equations with five unknown functions. For similar systems, and hence for quasi-linear equations, there are existence and uniqueness theorems for the Cauchy problem. This method can be applied without any significant changes to quasi-linear second-order systems

$$ a u _ {xx} ^ {j} + b u _ {xt} ^ {j} + c u _ {tt} ^ {j} + d ^ {j} = 0 ,\ \ j = 1, \dots, k , $$

where the coefficients depend on $ x , t $ and the functions $ u _ {j} $.

References

[1]	J. Glimm, "Solutions in the large for nonlinear hyperbolic systems of equations" Comm. Pure Appl. Math. , 18 (1965) pp. 697–715
[2]	A. Douglis, "Layering methods for nonlinear partial differential equations of first order" Ann. Inst. Fourier , 22 (1972) pp. 141–227
[3]	S.N. Kruzhkov, "Generalized solutions of the Cauchy problem in the large for nonlinear equations of first order" Soviet Math. Dokl. , 10 : 4 (1969) pp. 785–788 Dokl. Akad. Nauk SSSR , 187 : 1 (1969) pp. 29–32
[4]	N.N. Kuznetsov, "Global solutions of the Cauchy problem for quasi-linear first order equations in several space variables" Math. Notes , 2 : 4 (1967) pp. 733–739 Mat. Zametki , 2 : 4 (1967) pp. 401–410
[5]	R. Courant, P. Lax, "On nonlinear partial differential equations with two independent variables" Comm. Pure Appl. Math. , 2 (1949) pp. 255–273
[6]	E.D. Conway, J. Smoller, "Weak solution of the Cauchy problem for a multi-dimensional quasi-linear equation" Comm. Pure Appl. Math. , 19 (1966) pp. 95–105
[7]	P. Lax, "Nonlinear hyperbolic equations" Comm. Pure Appl. Math. , 6 (1953) pp. 231–258
[8]	H. Lewy, "Ueber das Anfangswertproblem einer hyperbolischen nichtlinearen partiellen Differentialgleichung zweiter Ordnung mit zwei unabhängigen Veränderlichen" Math. Ann. , 98 (1927) pp. 179–191
[9]	E. Levi, "Sur problema di Cauchy per le equazioni a caratteristiche reali e distinti" Rend. R. Acad. Lincei (5) , 17 (1908) pp. 331–339
[10]	J. Leray, "Hyperbolic differential equations" , Princeton Univ. Press (1953)
[11]	Da-tsin Lee, Wen-tzu Yu, "Some existence theorems for quasi-linear hyperbolic systems of partial differential equations in two independent variables I" Scientia Sinica , 13 : 4 (1964) pp. 529–562
[12]	O.A. Oleinik, "Discontinuous solutions of nonlinear differential equations" Uspekhi Mat. Nauk , 12 : 3 (1957) pp. 3–73 (In Russian)
[13]	I.G. Petrovskii, "Ueber das Cauchysche Problem für Systeme von partielle Differentialgleichungen" Mat. Sb. , 2 (44) (1937) pp. 815–870
[14]	B.L. Rozhdestvenskii, N.N. Yanenko, "Systems of quasilinear equations and their applications to gas dynamics" , Amer. Math. Soc. (1983) (Translated from Russian)
[15]	K. Friedrichs, "Nonlinear hyperbolic differential equations for functions of two independent variables" Amer. J. Math. , 70 (1948) pp. 555–589
[16]	P. Hartman, A. Winter, "On the hyperbolic partial differential equations" Amer. J. Math. , 74 (1952) pp. 834–864
[17]	J. Schauder, "Cauchysche Problem für partielle Differentialgleichungen erster Ordnung. Anwendung einiger sich auf die Absolutbeträge der Lösungen beziehenden Abschätzungen" Comment. Math. Helv. , 9 (1936) pp. 263–283

Comments

For applications see [a2]–[a3].

References

[a1]	A. Jeffrey, "Quasilinear hyperbolic systems and waves" , Pitman (1976)
[a2]	G.B. Whitham, "Linear and non-linear waves" , Wiley (1974)
[a3]	H.K. Rhee, R. Aris, N.R. Amundson, Philos. Transactions Roy. Soc. A , 267 (1970) pp. 419–455